dynamic analysis and fault detection of...

DYNAMIC ANALYSIS AND FAULT

DETECTION OF MULTI CRACKED

STRUCTURE UNDER MOVING MASS

USING INTELLIGENT METHODS

Shakti Prasanna Jena

Department of Mechanical Engineering

National Institute of Technology Rourkela

DYNAMIC ANALYSIS AND FAULT

DETECTION OF MULTI CRACKED

STRUCTURE UNDER MOVING MASS

USING INTELLIGENT METHODS

Thesis Submitted to the


National Institute of Technology, Rourkela

in partial fulfilment of the requirements

of the degree of

Doctor of Philosophy

in

Mechanical Engineering

by

Shakti Prasanna Jena

(Roll Number: 512ME123)

under the supervision of

Prof. Dayal R. Parhi

December 2016


National Institute of Technology Rourkela

i



December 7, 2016

Certificate of Examination

Roll Number: 512ME123

Name: Shakti Prasanna Jena

Title of Dissertation: Dynamic Analysis and Fault Detection of Multi Cracked Structure

under Moving Mass using Intelligent Methods

We, the below signed, after checking the dissertation mentioned above and the official

record book (s) of the student, hereby state our approval of the dissertation submitted in

partial fulfilment of the requirements of the degree of Doctor of Philosophy in Mechanical

Engineering at National Institute of Technology Rourkela. We are satisfied with the

volume, quality, correctness, and originality of the work.

---------------------------

Dayal R. Parhi

Principal Supervisor

--------------------------- ----- -----------------------------

S. Murugan R. Mazumder

Member (DSC) Examiner Member (DSC)

------------------------------- ------------------------------

S. K. Das K V Sai Srinadh

Member (DSC) Examiner External Examiner

------------------------------ ------------------------------

K. P. Maity S. S. Mahapatra

DSC Chairman Head of the Department

ii



Supervisor Certificate

This is to certify that the work presented in this thesis entitled “Dynamic Analysis and

Fault Detection of Multi Cracked Structure under Moving Mass using Intelligent

Methods” by “Shakti Prasanna Jena”, Roll Number 512ME123, is a record of original

research carried out by him under my supervision and guidance for the partial fulfilment

of the requirements of the degree of Doctor of Philosophy in the Department of

Mechanical Engineering, National Institute of Technology, Rourkela, Odisha, India.

To the best of my knowledge, neither this thesis nor any part of it has been submitted for

any degree or diploma to any institute or university in India or abroad.

--------------------------

December 7, 2016 Supervisor

(Dr Dayal R. Parhi )

Professor



Odisha, India

iii

Declaration of Originality

I, Shakti Prasanna Jena, Roll Number 512ME123 hereby declare that this dissertation

entitled “Dynamic Analysis and Fault Detection of Multi Cracked Structure under

Moving Mass using Intelligent Methods” represents my original work carried out as a

doctoral student of NIT, Rourkela and, to the best of my knowledge, it contains no

material previously published or written by another person, nor any material presented for

the award of any other degree or diploma of NIT, Rourkela or any other institution. Any

contribution made to this research by others, with whom I have worked at NIT, Rourkela

or elsewhere, is explicitly acknowledged in the dissertation. Works of other authors cited

in this dissertation have been duly acknowledged under the section ''Bibliography''. I have

also submitted my original research records to the scrutiny committee for evaluation of my

dissertation. I am fully aware that in case of any non-compliance detected in future, the

Senate of NIT, Rourkela may withdraw the degree awarded to me on the basis of the

present dissertation.

December 7, 2016 Shakti Prasanna Jena

NIT Rourkela

iv

Acknowledgements

Though only my name appears on the cover of this dissertation, many people have contributed to

its production. I owe my gratitude to all those people who have made this dissertation possible and

because of whom my doctorate experience has been one that I will cherish forever.

My deepest gratitude is to my supervisor, Prof. Dayal R. Parhi. I have been amazingly fortunate to

have an advisor who gave me the freedom to explore on my own and at the same time the

guidance to recover when my steps faltered. His patience and support helped me overcome many

crisis situations and finish this dissertation. I hope that one day I would become as good an advisor

to my students as he has been to me.

I am thankful to Prof. Animesh Biswas, Director of National Institute of Technology, for giving

me an opportunity to be a part of this institute of national importance and to work under the

supervision of Prof. Dayal R. Parhi. I am thankful to Prof. S. S. Mahapatra, Head of the

Department, Department of Mechanical Engineering, for his moral support and valuable

suggestions during the research work.

I express my gratitude to Prof. K.P. Maity (Chairman DSC) and DSC members for their indebted

help and valuable suggestions for accomplishment of dissertation.

I thank all the members of the Department of Mechanical Engineering, and the Institute,

who helped me in various ways towards the completion of my work.

I would like to thank all my friends (Durga, Madhu, Sidha and Smruti) and lab-mates

(Prases, Biplab, Kaku, Animesh, Subhasri, Sasmita, Anish, Yadoo, Chinu, Ghodki, Alok, Prabir

and Irshad) in Robotics lab for their encouragement and understanding. Their support and lots of

lovely memory with them can never be captured in words. My special thanks to Mr Prases

Mohanty helping me a lot and working in the laboratory for late nights. I want to especially

thankful Mr. Mahesh, for helping me in various ways throughout my Ph.D work.

I thank my parents, my grandmother, my uncles (Lalit & Bipin) and entire family members for

their unlimited support and strength. Without their dedication and dependability, I could not

have pursued my PhD degree at the NIT, Rourkela. I would like to give special thanks to my wife

(Reena) for her continuous supports and encouragements in my days of PhD work.

Last, but not the least, I praise the Almighty for giving me the strength during the research work.

December 7, 2016 Shakti Prasanna Jena

NIT Rourkela Roll Number: 512ME123

v

Abstract

The present thesis explores an inclusive research in the era of moving load dynamic

problems. The responses of vibrating structures due to the moving object and different

methodologies for damaged identification process have been investigated in this analogy.

The theoretical-numerical solutions of the multi-cracked structure with different end

conditions subjected to transit mass have been formulated. The Runge-Kutta fourth order

integration approach has been applied to determine the response of the structures

numerically. The effects of parameters like mass and speed of the traversing object, crack

locations, and depth on the response of the structures are investigated. The proposed

numerical method has been verified using FEA and experimental investigations. The novel

damage prediction processes are developed on the knowledge-based concepts of recurrent

neural networks (RNNs) and statistical process control (SPC) methods as inverse

approaches. The Jordan’s recurrent neural networks (JRNNs), Elman’s recurrent neural

network (ERNNs), the integrated approach of the JRNNs, and ERNNs, the autoregressive

(AR) process in the domain of SPC and the combined hybrid neuro-autoregressive process

have been developed to identify and quantify the faults in the structure. The accuracy and

exactness of each approach has been verified with experiments and FEA. The proposed

methods can be useful for the online condition monitoring of faulty cracks in structures.

Keywords: cracked beam, Runge-Kutta, RNNs and autoregressive.

vi

Content

Certificate of Examination ................................................................................................. i

Supervisor Certificate ....................................................................................................... ii

Declaration of Originality ................................................................................................ iii

Acknowledgement ............................................................................................................. iv

Abstract .............................................................................................................................. v

Contents ............................................................................................................................. vi

List of Figures ................................................................................................................... ix

List of Tables .................................................................................................................... xii

Nomenclatures ................................................................................................................ xiv

1 Introduction ................................................................................................................. 1

1.1 Motivation ............................................................................................................... 1

1.2 Objective of the Research Works ............................................................................ 2

1.3 Novelty of the Research Work ................................................................................ 3

1.4 Contributions ........................................................................................................... 4

1.5 Overview of the thesis ............................................................................................. 4

2 Literature Review ........................................................................................................ 6

2.1 Introduction ............................................................................................................. 6

2.2 Response analysis of structure ................................................................................. 6

2.2.1 Classical methods ........................................................................................... 6

2.2.2 Finite element analysis/method for response analysis of structure ............... 12

2.3 Damage detection .................................................................................................. 15

2.3.1 Classical/FEA based Methods for damage detection in structure ................ 15

2.3.2 AI techniques based methods for fault detection in structure ....................... 22

2.3.2.1 Genetic Algorithm based methods for crack detection in structure . 22

2.3.2.2 Neural network based methods for crack detection in structure ...... 23

2.3.2.3 Recurrent neural network based methods for crack detection in

structure ........................................................................................................ 27

2.4 Statistical based method for fault identification in structure ................................. 28

2.5 Miscellaneous methods .......................................................................................... 29

2.6 Summary ................................................................................................................ 30

vii

3 Theoretical-Numerical Analysis of Multi-cracked Structures Subjected to

Moving Mass .............................................................................................................. 32

3.1 Introduction ........................................................................................................... 32

3.2 The problem Description ....................................................................................... 32

3.3 The Problem Formulation ...................................................................................... 33

3.4 Analysis of Cracked Structures Subjected to Moving Mass ................................. 33

3.5 Numerical Formulation of Cracked Cantilever Beam under a Moving Mass ....... 37

3.6 Theoretical-Numerical Solution of Cracked Simply Supported Structure under a

Moving Mass ......................................................................................................... 45

3.7 Theoretical-Numerical Solution of Cracked Fixed-Fixed Beam under a Moving

Mass ....................................................................................................................... 53

3.8 Identification of cracks from the measured dynamic response of structures ........ 61

3.9 Comparison of Results of Theoretical-Numerical Experimental analysis for the

Response of Structures ........................................................................................... 64

3.10 Discussions and Summary ................................................................................... 67

4 Finite Element Analysis of Cracked Structures Subjected to Moving Mass ....... 70

4.1 Introduction ........................................................................................................... 70

4.2 Method for FEA of moving mass-structure using ANSYS ................................... 70

4.3 Steps involving 'The full method' transient dynamic analysis in ANSYS ............ 72

4.4 Response analysis of cracked structures under moving mass using ANSYS ........ 72

4.5 Discussion and Summary ...................................................................................... 84

5 Application of Recurrent Neural Networks for Damage Identification in

Structures Under Moving Mass ............................................................................... 86

5.1 Introduction ........................................................................................................... 86

5.2 Overview of neural networks ................................................................................ 87

5.2.1 Feed forward neural networks ...................................................................... 87

5.2.2 Recurrent neural networks (RNNs) .............................................................. 88

5.3 Use of Levenberg-Marquardt back propagation method for RNN ........................ 92

5.3.1 Steps for the organization of the training procedure using L.M algorithm .. 93

5.4 Application of rule-based modified JRNNs for damage identification in

structure under moving mass ................................................................................. 94

5.5 Application of rule-based modified FRNNs for damage detection in structure

subjected to moving mass ...................................................................................... 98

5.6 Application of rule-based modified hybridized JRNNs and ERNNs for

viii

for multiple damage detection in structure subjected to moving mass ................. 101

5.7 Discussion and Summary .................................................................................... 113

6 Application of Statistics Based Damage Identification Procedure for

Structures under Transit Mass .............................................................................. 114

6.1 Introduction ......................................................................................................... 114

6.2 Overview of Statistical Process Control (SPC) method ...................................... 115

6.3 Construction and analysis of control chart .......................................................... 115

6.4 Overview of Autoregressive model ..................................................................... 117

6.5 Application of auto regressive (AR) model based method for damage detection

in structures subjected to traversing mass ............................................................ 119


7 Combined Hybrid Neuro-Autoregressive Model for Fault Detection in Beam

Structures Subjected to Moving Mass ................................................................... 132

7.1 Introduction ......................................................................................................... 132

7.2 Development of combined hybrid neuro-autoregressive model for damage

detection in beam type structures subjected to moving mass ............................... 133


8 Experimental Analysis of Damaged Structures Subjected to Transit Mass ...... 139

8.1 Introduction ........................................................................................................ 139

8.2 Experimental Procedure ...................................................................................... 139


9 Results and Discussion ............................................................................................ 145

9.1 Introduction ........................................................................................................ 145

9.2 Analysis and results of different adopted methods .............................................. 145

9.3 Summary .............................................................................................................. 149

10 Conclusions and Suggestion for Further Research .............................................. 150

10.1 Introduction ...................................................................................................... 150

10.2 Contributions ..................................................................................................... 150

10.3 Conclusions ....................................................................................................... 151

10.4 Recommendation for future study ..................................................................... 152

Appendix ......................................................................................................................... 153

Bibliography ................................................................................................................... 156

Dissemination ................................................................................................................. 176

Vitae ............................................................................................................................. 178

ix

List of Figures

3.1 Multi- cracked cantilever beam under moving mass .......................................... 34

3.2 For undamaged beam for 438 /v cm s ............................................................... 38

3.3 For undamaged beam for 573 /v cm s ............................................................... 38

3.4 For 1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s ............................... 39

3.5 For 1,2,3 1,2,3573 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s ................................ 39

3.6 For 1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.5,0.65,0.85v cm s ................................. 40

3.7 For 1,2,3 1,2,3573 / , 0.6,0.25,0.45. 0.5,0.65,0.85v cm s ................................. 40

3.8 For 1,2,3 1,2,3438 / , 0.3,0.55,0.4. 0.25,0.45,0.65v cm s ................................. 41

3.9 For 1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.25,0.45,0.65v cm s .................................. 41

3.10 For 1,2,3 1,2,3438 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s ................................. 42

3.11 For 1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s ................................. 42

3.12 3-D Graph for time mass deflection for

1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s ........................................ 43


1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s ………………………………….43

3.14 3-D Graph for time speed deflection for

1,2,3 1,2,32 , 0.6,0.25,0.45. 0.25,0.45,0.65M kg ........................................... 44

3.15 3-D Graph for position time deflection for

1,2,3 1,2,32 , 573 / , 0.6,0.25,0.45. 0.25,0.45,0.65M kg v cm s ....................... 44

3.16 Multi-cracked simply supported beam under moving mass .............................. 45

3.17 For undamaged beam for 438 /v cm s ............................................................. 46

3.18 For undamaged beam for 573 /v m s ............................................................... 46

3.19 For 1,2,3 1,2,3438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s ........................... 47

3.20 For 1,2,3 1,2,3573 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s ........................... 47

3.21 For 1,2,3 1,2,3438 / , 0.35,0.45,0.55. 0.2857,0.5,0.7143.v cm s ..................... 48

3.22 For 1,2,3 1,2,3573 / , 0.35,0.45,0.55. 0.2857,0.5,0.7143.v cm s ..................... 48

3.23 For 1,2,3 1,2,3438 / , 0.2,0.3,0.4. 0.1786,0.3571,0.5714.v cm s ...................... 49

3.24 For 1,2,3 1,2,3573 / , 0.2,0.3,0.4. 0.1786,0.3571,0.5714.v cm s …….. ........... 49

3.25 For 1,2,3 1,2,3438 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s ................ 50

3.26 For 1,2,3 1,2,3573 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s ............ …50

3.27 3-D Graph for time ~mass ~deflection for

x

1,2,3 1,2,3 573 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714v cm s .... ……………51


1,2,3 1,2,3 2 , 0.35,0.45,0.55. 0.1786,0.3571,0.5714M kg …………………………...51


1,2,3 1,2,3 438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143v cm s ................................... 52

3.30 3-D Graph for position ~ time ~ deflection for

1,2,3 1,2,3 1 , 438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143M kg v cm s .................. …52

3.31 Multi-cracked fixed-fixed beam subjected to moving mass .............................. 53

3.32 For undamaged beam for 512 /v cm s …………………………………….......54

3.33 For undamaged beam for ................................................................................... 54

3.34 For 1,2,3 1,2,3512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357v cm s ........... …….55

3.35 For 1,2,3 1,2,3617 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357v cm s .................... 55

3.36 For 1,2,3 1,2,3512 / , 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s ....................... 56

3.37 For 1,2,3 1,2,3617 / , 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s ....................... 56

3.38 For 1,2,3 1,2,3512 / , 0.3,0.5,0.55. 0.1429,0.3214,0.5357v cm s ...................... 57

3.39 For 1,2,3 1,2,3617 / , 0.3,0.5,0.55. 0.1429,0.3214,0.5357v cm s ...................... 57

3.40 For 1,2,3 1,2,3512 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143v cm s .......................... 58

3.41 For 1,2,3 1,2,3617 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143v cm s .......................... 58


1,2,3 1,2,3 617 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143.v cm s .............................. 59


1,2,3 1,2,3 2 , 0.3,0.5,0.55. 0.25,0.4286,0.7143.M kg ...................................... 59


1,2,3 1,2,3 512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357.v cm s ......................... 60

3.45 3-D Graph for Position time deflection for 2 ,M kg

1,2,3 1,2,3 512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357.v cm s ......................... 60

3.46 Detection of cracks for cantilever beam for 1,2,3 0.5,0.65,0.85 ...................... 61

3.47 Detection of cracks for simply supported beam for

1,2,3 0.2857,0.5,0.7143. ................................................................................ ....62

3.48 Detection of cracks for fixed-fixed beam for 1,2,3 0.25,0.4286,0.7143. …..….62

3.49a Magnified view of crack for 0.5 ………………………...…………………63

3.49b Magnified view of crack for 0.65 ………………………...…………………63

3.49c Magnified view of crack for 0.85 ..………………………….…………….64

4.1 Free body diagram of vibrating system……………………………………...…71

xi

4.2 Transit mass-structure interaction of cracked cantilever beam for

1,2,3 1,2,30.3,0.55,0.4. 0.5,0.65,0.85. M=2 kg…………..…………………....73

4.3 Magnified view of crack for α=0.55 ......................................................... …….73

4.4a Second mode shape of cantilever structure .............................................. …….74

4.4b Third mode shape of cantilever structure ................................................. …….75

4.5 Schematic view of transient structural model for cracked cantilever beam…...75

4.6 For cracked cantilever beam for 1,2,3 1,2,30.3,0.55,0.4. 0.25,0.65,0.85 ....... 77

4.7a Second mode shape of simply supported beam ................................................. 78

4.7b Third mode shape of simply supported beam .................................................... 79

4.8 For cracked simply supported beam for, 2 , 438 /M kg v cm s

1,2,3 1,2,30.35,0.45,0.55. 0.2857,0.5,0.7143 .................................................. 81

4.9a Second mode shape fixed-fixed beam ............................................................... 81

4.9b Third mode shape fixed-fixed beam .................................................................. 82

4.10 For cracked fixed-fixed beam for 2 , 617 /M kg v cm s ,

1,2,3 1,2,30.2,0.35,0.45. 0.1429,0.3214,0.5357. ............................................ 83

5.1 Simplified NN model with feed forward networks ................................... …….87

5.2 Architecture of a RNN model ............................................................................. 88

5.3 Simple Architecture of JRNN model ................................................................. 90

5.4 Simple architecture of ERNN model .............................................................. …90

5.5 Simple architecture of HRNN model ........................................................ …….91

5.6 Architecture of modified JRNN model ..................................................... …….95

5.7 Architecture of modified ERNN model .................................................... …….99

5.8 Hybridized architecture of modified JRNN and ERNN models ...................... 102

5.9 Plot of graph of iterations vs. sum square errors for RNNs methods model .... 104

6.1 Architecture of control chart ................................................................... …….116

6.2 Representation of a probability process as the outputs from a liner filter…….117

6.3 Data analysis in SPSS windows ....................................................................... 120

6.4a Control chart for cantilever beam .................................................................... 120

6.4b Control chart for simply supported beam ........................................................ 121

6.4c Control chart for fixed-fixed beam .................................................................. 121

7.1 Architecture of Hybrid neuro autoregressive model ....................................... 133

8.1 Experimental set up for cantilever beam ..................................................... …140

8.2 Experimental set up for simply supported beam .................................... …….141

8.3 Experimental set up for fixed-fixed beam .............................................. …….141

8.4 Ultrasonic sensor ..................................................................................... …….142

8.5 Micro-controller Aurdino ............................................................................... 142

8.6 Damaged portion of beam .............................................................................. 143

8.7 Variac........................................................................................................144

8.8 Bread board...............................................................................................144

xii

9.1 Comparison of results among different damage detection methods .............. 149

A1 Crack analysis on the vibration characteristics of cantilever beam .............. …153

List of Tables

3.1 Comparison of results between experiment and numerical for cracked cantilever

beam for 1,2,3 1,2,3438 / . 0.6,0.25,0.45. 0.25,0.45,0.65.v cm s ...................... 65

3.2 Comparison of results between experiment and numerical for cracked simply

supported beam for 1,2,3 1,2,3573 / . 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s .... 66

3.3 Comparison of results between experiment and numerical for cracked

fixed-fixed beam for 1,2,3 1,2,3617 / . 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s ..66

4.1 Frequencies ratios of damaged cantilever beam .................................................. 74

4.2 Comparison of results between experiment and FEA for cracked cantilever beam

for 1,2,3 1,2,3573 / . 0.3,0.55,0.4. 0.5,0.65,0.85.v cm s .................................... 76

4.3 Comparison of results among experiment, FEA and numerical for cracked

cantilever beam for M=1kg,v=438cm/s, 1,2,3 1,2,30.3,0.55,0.4. 0.5,0.65,0.85 ..... 77

4.4 Frequencies ratios of damaged simply supported beam ...................................... 79

4.5 Comparison of results between experiment and FEA for cracked simply

supported beam for 1,2,3 1,2,3438 / . 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s .... 79

4.6 Comparison of results among experiment, FEA and numerical for cracked simply

supported beam for M=1kg, v=573cm/s, 1,2,3 1,2,30.35,0.45,0.55. 0.1876,0.3571,0.5714. ...80

4.7 Frequencies ratios of damaged fixed-fixed beam…………………………….…80

4.8 Comparison of results between experiment and FEA for cracked fixed-fixed

beam for 1,2,3 1,2,3512 / . 0.3,0.5,0.55. 0.25,0.4286,0.7143.v cm s ................. 82

4.9 Comparison of results among experiment, FEA and numerical for cracked

fixed-fixed beam for M=2kg, v=617cm/s, 1,2,3 1,2,30.3,0.5,0.55. 0.25,0.4286,0.7143. ..83

5.1 Test patterns to the RNN model for cracked cantilever beam ........................... 104

5.2(a, b) Comparison of results between experiments and different RNNs method

for prediction of relative crack locations (cantilever beam)………………105-106

5.3(a,b) Comparison of results between experiments and different RNNs method

for prediction of relative crack locations (simply supported beam) ............ 107-108

5.4(a,b) Comparison of results between experiments and different RNNs method

for prediction of relative crack locations (fixed-fixed beam) ....................... 109-110

5.5(a,b) Comparison of results among various methods for prediction of relative

crack locations (fixed-fixed beam) ............................................................. 111-112

xiii

6.1(a,b) Comparison of results among FEA, Theoretical and SPC methods for

estimation of relative crack locations (cantilever beam) ............................ 125-126

6.2(a,b) Comparison of results among FEA, Theoretical and SPC methods for

estimation of relative crack locations (simply supported beam) ......... 126-127

6.3(a,b) Comparison of results among Experimental, FEA, Theoretical and

SPC methods for estimation of relative crack locations (fixed-fixed

beam)…………………………………………………………………128-129

7.1(a,b) Comparison of results among FEA, Theoretical and hybrid approach

of RNNs, and SPC methods for prediction of relative crack locations

(cantilever beam) ........................................................................................ 135

7.2(a,b) Comparison of results among Experimental, FEA, Theoretical and

hybrid approach of RNNs, and SPC methods for prediction of relative

crack locations (simply supported beam) ................................................... 136

7.3(a,b) Comparison of results among FEA, Theoretical and hybrid approach

of RNNs, and SPC methods for prediction of relative crack locations

(fixed-fixed beam) ....................................................................................... 137

8.1 Specification of different components of experimental set up .......................... 143

xiv

Nomenclature

( )u x u = Deflection due to longitudinal vibration of the beam. ( )y x y = Deflection due to transverse vibration of the beam. ( )I x I = Moment of inertia of the beam.

( )m x m = Beam mass per unit span length.

g = gravitational constant

L =Length of the beam.

( )F t The interactive force between the moving mass and the structure.

1 2 3 1,2,3, ,L L L L = Position of the first, second and third cracks at the left end of the

cracked structures respectively.

1,2,31,2,3

L

L =Relative crack Positions from the fixed end.

1 2 3 1,2,3, ,d d d d = Depth of the first, second and third cracks respectively.

1,2,31,2,3

dH

= Relative crack depth

M = Mass of the moving mass

v =Moving speed of the transit mass.

vt = Position of the moving mass at any instant time‘t’.

h = The point of attention where the beam deflection is to be determined.

( , )r x t Standard loading conditions.

ϐ = Sheer force, = Bending moment.

δ=Dirac delta function.

( )n x = Eigen functions of the beam.

( )nQ t = Amplitude functions of the beam.

n = Number modes of vibrations.

[ ]tM x -Inertial force

[ ]tC x -Damping force

[ ]tK x -Stiffness force,

xv

( )f t -Applied force.

ρ- Density of the beam.

E- Young’s modulus of elasticity.

υ- Poisson’s ratio

k- Stiffness

G – Rigidity of the beam.

f(.) and g(.) - Activation functions in the hidden and output layers respectively.

w- Synaptic weights.

W-Input values to the neural networks.

- Output values from the neural networks.

1Z - The delay units of the neural networks.

J - The Jacobian matrix.

ξ- The combination coefficient.

ν - The step size or training constant.

X - Identity matrix.

e - The error value.

- The error function.

- The self-recurrent values of the each node.

- The learning rate of the neural network.

- The mean. w - The standard deviation.

( )b - Autoregressive operator.

s - The sample size.

ta - Shock or white noise, c- Order of the autoregressive process

- Co-efficient of autoregressive process.

- The covariance matrix.

V - The variance.

-The probability density function.

s - The statistical moment.

tz - The linear filter.

1

Chapter 1

INTRODUCTION

For most part, structures in civil, mechanical and aerospace engineering applications are

subjected to space and time varying masses or loads. Moving masses or loads have

significant consequence on the dynamic characteristics of the structures. Dynamic analysis

of structures under the action of moving mass is the traditional subject of research. The

problems of the dynamic response of structures subjected to moving mass have been

investigated by numerous scientists, engineers, researchers and mathematicians for last

few decades. But the first incident of moving mass problem came in 1847; the collapse of

the Chester Bridge, England [Atkin and Mofid, 3], which eventually led heartbreaking

hammering of human beings. The solutions of the moving mass problems draw attentions

of several scientists, engineers, researchers and mathematicians to investigate the moving

mass problem. Initially, the solutions for the moving mass problems are formulated on the

basic assumption, and more often, ignoring the mass of the traversing object. Later, the

problem involves the inclusions of inertial, rotary, shearing and damping effects. But due

to the existence of different types of cracks, increased mass and speed of the moving

object, and structural boundary condition, the problems become more complicated.

Therefore, enhanced learning mechanism is necessary to investigate the response of

structures, early detection of crack locations and severities to improve the service periods

of structure, mass, human beings and better economic growth.

This Chapter describes the background and motivation, objectives and scopes, and

overviews of the present thesis.

1.1 Motivation

The applications of engineering structures under moving mass have gained enormous

importance in the field of aerospace, buildings, defence, cranes, bridges, railways

engineering, etc. The dynamic behaviour of moving mass-structural systems has been

characterised by geometrical and mathematical modelling along with the consideration of

initial conditions, different end conditions, geometric characters, material properties, basic

assumptions and a set of coupled partial differential equations. The occurrence of faults in

Chapter 1 Introduction

2

engineering structures may be unavoidable in spite of our greatest efforts. The faults may

come concerning cracks, fatigue, enhanced moving speed of the mass, enhancement of

mass, mechanised progression, erosion, etc. The faults are induced in the structure due to

several reasons such as metallurgical defects, manufacturing defects, environmental

defects and mechanical defects. So it is important to study the integrity and consistency of

structures for better service periods.

Various analyses, experiments and methodologies developed by engineers, scientists and

researchers, provide the inspiration and the support for the innovative research. The

existence of faults deteriorates the behaviour of the structures up to a certain degree. So it

is required for the early detection of faults for sustaining the stability and existence of

structures. There are several methods developed by various researchers for fault diagnosis

of structures under moving mass. Most of the methods (gamma radiation, ultrasonic

testing, X-ray testing, magnetic particle and etc.) are limited to local damage isolation, less

sensitive and costly. But vibration based damage isolation methods are global and highly

sensitive. So Artificial Intelligence (AI) techniques as damage detector are developed by

changing the dynamic properties and vibration signatures of structures by many

researchers which are global and cost-efficient. Here some rule-based Recurrent Neural

Network methods, statistics based method along with some novel hybridised methods are

proposed to identify, estimate and quantify the extent and severities of cracks. The

responses of different types of structures under traversing mass with multiple cracks are

also determined using analytical- numerical solutions along with FEA and experimental

verifications.

1.2 Objectives of the Research Works

The effect of moving load over a structure is the fundamental problem in structural

dynamics. The studies of response and the parameters influencing the responses of

structures to moving mass are essential for the integrity and health monitoring of

structures. The primary objectivities of the current thesis are to determine the response of

the multi-cracked structure under moving mass with different boundary states using

theoretical, FEA and experimental analyses, and to develop reverse methodology on crack

detection. The specific objectives of the present thesis are:


3

1) The presentation of simple and sensible methods for determining the response of multi-

cracked structures with different end conditions subjected to a moving mass.

2) The formulation of solution to the problem in analytical-numerical form.

3) The verification to the solution of the problem obtained from the analytical-numerical

method using FEA and experimental analysis.

4) The analyses of the significance of various parameters and its influences on the

response of the structures.

5) The development of damage isolation procedure based on AI technique using the

change in dynamic response of the structure.

6) The development of rule-based RNNs (Jordan and Elman) for damage isolation.

7) The hybridisation of Jordan’s and Elman’s RNNs for damage isolation.

8) The development of a novel damage isolation method based on statistical data analysis.

9) The development of a novel hybridised method using the rule-based RNNs and

statistical data analysis for damage identification.

1.3 Novelty of the Research Work

The present research work is concerned with the determination of the response of

structures under the influence of moving mass and the development of novel damage

isolation procedures based on improved vibration-based damage identification method.

The novelty of the proposed work is restricted to the subsequent areas:

1) The investigation of the theoretical-numerical method along with FEA and

experimental verifications.

2) The consideration of different parameters and their influences on the response of the

structures subjected to moving mass.

3) The investigation of novel AI techniques for damage isolation in structures based on the

concepts of RNNs (Jordan’s, Elman’s and hybridisation of Jordan’s, and Elman’s RNNs),

statistical process control method and vibration induced data.

To the best of Author’s knowledge, the applications of knowledge-based RNNs methods

for damage detection in structures are not reported previously. The damage detection

method using the displacement response of structure in statistical process control analysis

is not cited earlier. The application of the combined hybrid neuro-autoregressive process

for structural damage detection is scanty.


4

1.4 Contributions

The most important role of this research work is the improvement and validation of novel

vibration-based damage isolation methods. The vibration-based damage isolation methods

include the formulation of rule-based RNNs (Jordan and Elman) method. This method is

also based on statistical data analysis and the integration of the hybridised RNNs, and

statistical data analysis method. The proposed methods have been applied to improve the

original damage isolation methods for better prediction of the damage identification,

localisation and quantification of damage severities. All the methods have been verified

with numerical, FEA and experimental tests.

1.5 Overview of the thesis

The entire thesis includes ten Chapters. Brief descriptions of all the Chapters are as

follows:

Chapter-1 presents the introductory knowledge on moving load problems. This Chapter

includes the motivation behind this study, research aim and objectives, the scope of the

research and summarisation of the present work.

Chapter-2 explains about research works carried out by various researchers. This Chapter

presents various methodologies, analyses and experiments on moving load problem. This

Chapter gives the insufficiency in literature reviews and formulates the novelty of the

present study.

Chapter-3 formulates an analytical-numerical method for the solution of different types

of structures under moving mass.

Chapter-4 carries out FEA for the solution of the moving mass problem on structure.

Chapter-5 formulates the fault analysis of multiple cracked structures subjected to

moving mass using different types of rule-based RNNs.

Chapter-6 explains the formulation of statistics based damage isolation method for the

multi-cracked structure under moving mass.

Chapter-7 presents the integrated approach of the hybridised RNN method and the rule-

based statistics based method.

Chapter-8 conducts laboratory tests for different types of structures under moving mass.

Chapter-9 explains about the reviews of all the methodologies made in the thesis and

shows the comparison of results.


5

Chapter-10 discusses conclusions and scope of future work to be carried out.

6

Chapter 2

Literature Review

2.1 Introduction

This Chapter discusses the innovative works reported by various researchers in the field of

moving load structure interaction dynamics and its applications in various engineering,

transportation and infrastructure industries. Numerous techniques have been proposed to

determine the response and to diagnose faults in the structure under moving mass. Several

significant works of researchers which have received great attentions are explained briefly

in this Chapter.

2.2 Response analysis of structure

2.2.1 Classical methods

Sridharan and Mallik [1] carried out a numerical analysis to study the vibrational effects

beams under moving load. They employed the Wilson-θ method to analyze the response

of the structure. Siddiqui et al. [2] have investigated the dynamic response of a cracked

cantilever structure under a transit mass with internal resonance conditions. The response

of the moving mass-structure has been determined using the Rayleigh-Ritz method and the

perturbation method of multiple scales to obtain approximate solution. A theoretical-

numerical method has been developed by Akin and Mofid [3] to carry out the response of

the structure under moving mass using Runge-Kutta method. They have compared the

results from numerical analysis with finite element analysis (FEA) and found good

agreements. Stanisic and Hardin [4] have developed a theory to describe the response of

the structure under a random number of traversing masses using Fourier’s analysis.

Olsson [5] discussed the dynamic analysis of a simply supported structure under a

constant moving force travelling at constant speed. He has presented analytical and finite

element solution to this simply supported beam problem. Mofid and Akin [6] have

presented a novel experimental and an inexact method to calculate the response of the

structure under a moving load with respect to time. This inexact method is applicable for

wide range of structure with various end conditions. Kwon et al. [7] examined the analysis

Chapter 2 Literature Review

7

of vibration and control of bridge structure subjected to moving load using Tuned Mass

Damper (TMD) method. Mahmoud and Abouzaid [8] presented an iterative modal

analysis technique to examine the consequence of transverse crack on the response of a

simply supported structure subjected to travelling mass. They have explained the exact

effects of cracks and mass depending on the various parameters like traversing time,

moving speed, the location of cracks and crack types, etc.

Li et al. [9] have extracted the natural frequencies of railway girder bridges subjected to

moving load by using the eigenvalues solution of the moving load-bridge interaction

dynamics. Bilello et al. [10] conducted experimental verifications of a simply supported

highway bridge under a moving vehicle. They have applied the theory of structural model

to enlarge scale model experiment and to study the response of the bridge structure.

Bilello and Bergman [11] performed a theoretical investigation along with experimental

verification for the analysis of a cracked beam under travelling mass. Yang et al. [12]

presented a study to extort the fundamental frequency of a bridge structure by means of

the dynamic response of the moving vehicle across the bridge. Majka and Hartnett [13]

developed a proficient numerical model to investigate the consequence of different

parameters affecting the response of railway bridge structure. It has been observed that

parameters like the ratio of the train to bridge frequency, the speed of the train, span ratio,

mass and bridge damping are influencing the response of the structure.

Garinei and Risitano [14] have conducted a brief analysis of the conditions occurring from

the mutual loads spread over a single axle and mutual loads spread over a series of

equidistant loads of a railway bridge structure under moving load. Yang and Chang [15]

have conducted parametric studies to extract the bridge frequency indirectly from a

passing vehicle. Dehestani et al. [16] have developed a theoretical analysis followed by

computational analysis to study the response structure under moving mass with various

boundary conditions. The critical influential speed has been introduced in the analysis and

calculated numerically for different types of structures at various boundary conditions. Liu

et al. [17] have investigated the conditions of train-bridge interaction dynamics to study

the vibration induced due to the train-bridge interaction. The consequences of the train-

bridge interaction dynamics on the bridge response has been investigated at the time of the

passage of the train. The influence of different parameters like critical speed, the

frequency ratio of vehicle to bridge, vehicle model types and high modal damping value

have been investigated on the response of the bridge during the passage of the vehicle.


8

Siringoringo and Fujino [18] have carried out an analytical and experimental study to

approximate the bridge fundamental frequency indirectly from the dynamic response of

traversing vehicle. The vehicle load with pulsation sensor has been used to carry out

periodic measurement over various bridge structures and approximated their fundamental

frequencies. Xia et al. [19] investigated an analytical expression along with numerical

simulation and experimental verifications to investigate the resonance mechanism and

conditions of train-bridge structure. They observed that the resonance of the vehicle was

induced due to the periodical arrangement of bridge span and deflection. Majkaa and

Hartnett [20] have carried out a study to examine the dynamic effects induced by service

train, the significance of track randomness and bridge skewness. The extent of the

research reached at the observation that the significance of random track irregularities had

a small impact on the dynamic magnification factors and bridge acceleration, where as

track irregularities had the greater impact on bridge response.

A theoretical and experimental solution are carried out to explore the dynamic behaviour

of ground vibration induced due to a high-speed railway train, train on bridges and trains

in tunnels by Ju et al. [21]. The results indicated that the ground vibration induced due to

the train at the train load dominant frequencies are considerably great for both subsonic

and supersonic speeds of the train. Yoon et al. [22] have formulated a theoretical and

experimental analysis to examine the free vibration characteristics of a double cracked

simple supported Euler-Bernoulli beam with open cracks. The consequences of crack

depth and location on the natural frequency of the simple cracked beam were examined.

Mahmoud [23] has presented a method to calculate the stress intensity factor of a cracked

beam under travelling load with single and double edge cracks. Modal analysis method

was applied to calculate the equivalent load on the cracked structure and stress intensity

factor was calculated using the concepts of linear fracture mechanics. It has been observed

that stress intensity factor depends on moving mass speed, time, location and size of the

crack. Michaltsos et al. [24] have discussed the significance of traversing load and other

constraints on the response of a simply supported structure under moving mass. Mahmoud

[25] has described the response of a cracked, undamped simply supported beam under

traversing mass in the presence of a transverse crack. The effects of cracks on the response

of the simple beam were investigated through numerical solution.

Lin and Chang [26] obtained the response of a damaged cantilever beam along a

concentrated traversing load using the theoretical transfer matrix method. In the later part,


9

the forced response of the damaged structure was found out by using modal expansion

theory using the calculated eigenfunctions. Ouyang [27] carried out a tutorial on moving

load dynamic problems. Numerous types of essential concept related to structural dynamic

problems are explained in this study. Using the discrete element technique and finite

element method, Ariaei et al. [28] proposed a theoretical as well as calculation method to

obtain the response of a damaged structure under moving mass with open and breathing

cracks. The consequence of crack on the resonance condition of the structure was also

inspected. It was observed that crack can alter the critical speed leading to the resonance

of the structure. Azam et al. [29] have studied the response of a Timoshenko beam under

traversing and sprung mass structures. From various numerical examples, it was seen that

the deflection obtained due to moving mass system are higher than that of moving sprung

systems.

Michaltsos and Kounadis [30] have examined the consequence of centripetal, and Coriolis

forces on the response of a light girder bridge subjected to traversing load. Pala and Reis

[31-32] have investigated the significance of inertial, Coriolis, and centripetal, forces on

the dynamic response of cracked structures subjected to a traversing load with a single

crack. The response of the cracked structure was calculated using Duhamel integral. It was

concluded that the mass and speed of the traversing load influence the inertial, centripetal,

and Coriolis forces. Shi et al. [33] have developed a theoretical vehicle-bridge model to

investigate dynamic behaviour of slab bridges at various span lengths at different vehicle

speeds and road surfaces. Using finite deference method, Esmailzadeh and Ghorashi [34]

have studied the vibration of a Timoshenko beam under a partially dispersed moving load.

The response of the beam, bending moment and distribution of shear force are determined

in this analysis. Lee [35] presented the Lagrangian and the assumed mode approach for

analyzing the dynamic response of a Timoshenko beam. The study was carried out at

constant mass, speed and the beam slenderness ratio. Khalily [36] have studied the

dynamic behaviour of cantilever beam under moving mass. Mofid and Shadnam [37] have

developed an inexact method to describe the response of structures with internal hinges

and various end states under a traversing mass. Using the combined effect of finite

element and finite difference method, Cifuentes [38] has determined the response of a

beam vibrated by traversing mass. The methods used in the analysis are based on

Lagrange Multiplier. Nikkhoo et al. [39] have employed the most favourable control


10

algorithm with a time changing gain matrix with displacement speed feedback to examine

the response of structure excited by moving mass.

Grant [40] has analysed the consequences of rotary inertial and shearing deformation on

the transverse vibration of a Timoshenko beam under a concentrated mass. Thambiratnam

and Zhuge [41] have developed a simple method for the dynamic analysis of structures on

elastic foundation due to the effects of moving load. The effects of moving mass speed,

the dynamic magnification on deflection, stresses and the foundation stiffness on the

response of the structure are determined. Rao [42] has determined the dynamic behaviour

of Euler-Bernoulli beam under a traversing load using mode superposition method. The

method of multiple scales was applied to solve the equation of motion of the time varying

mass systems. Using discrete element technique, the dynamic behaviour of Timoshenko

beam vibrated by travelling load was studied by Yavari et al. [43]. The response of a

double -beam structure has been studied under constant moving mass speed by Abu-Hilal

[44]. The effects of the traversing mass speed, the elasticity of the viscoelastic level and

damping of the beam are investigated on the dynamic responses of the beams. Wang and

Zhang [45] have investigated the vibration analysis of a guideway due to the effects of a

moving maglev vehicle.

Yang et al. [46] have presented a methodical study on the free and forced vibration of a

non-homogeneous beam subjected to axial force and traversing mass with the presence of

open edge cracks. Yan et al. [47] have studied the dynamic behaviour of functionally

graded beams on elastic foundation with open edge crack carrying moving mass.

Parametric studies with different boundary conditions of structures are conducted to know

the importance of crack depth, crack location, gradient of material properties, slenderness

ratio of the beam, moving speed and stiffness of foundation. Shafiei and Khaji [48] have

proposed a theoretical solution to study the free and forced vibration of a Timoshenko

beam with multiple open edge cracks under a moving concentrated load. They have

determined forced vibration response of the structure using the method of modal

expansion. Suzuki [49] investigated the consequence of acceleration on the dynamic

response of finite beam subjected to traversing load. Employing double Laplace

transformation, Hamada [50] determined the response of a damped Euler-Bernoulli simply

supported beam under the action of traversing concentrated force. Lee and Ng [51] have

studied the vibration response of a beam carrying a moving mass on one surface and

having a single crack on opposite side. Ichikawa et al. [52] have explored the dynamic


11

behaviour of continuous structure carrying moving load using the concept of

eigenfunction expansion method. The influences of inertia and speed of moving load on

the response of the beam are described. Wu et al. [53] have employed both finite element

and analytical methods to study the response of a clamped-clamped beam under a

travelling mass primarily. The objective of this work was to predict the response of

moving crane structure using the above methods. Mallik et al. [54] have investigated the

steady-state response of an infinite beam on elastic support subjected to a traversing mass.

Aydin [55] have explained the characteristics of vibration of Euler-Bernoulli beams with

multiple cracks under the action of an axial load at numerous end conditions of the

structure. The effects of damages on buckling, edge cracks and axial loading on

eigenfrequencies and support conditions are discussed.

Sieniawska et al. [56] have detected the flexural stiffness of structure using the response of

traversing load. Yan and Yang [57] studied the flexural vibration of functionally graded

beams in the presence of open edge cracks under the action of both axial compressive

force and concentrated transverse load traversing along the beam. Using spectral element

method in time domain, Chen et al. [58] have presented the dynamic behaviour of

Timoshenko beam excited by an accelerating mass. Zarfam and Khaloo [59] have

investigated the response of structure on elastic foundation excited by traversing vehicle

and lateral vibration. Johansson et al. [60] have proposed a closed form solution to

examine the vibration analysis of a multi-span bridge structure carrying moving load. The

objective of the work was to carry out the vibration analysis of stepped beam under

constant traversing mass. Lou and Au [61] have developed finite element formulae to

evaluate the internal forces, shear forces and bending moment of Euler-Bernoulli beam

subjected to travelling mass. The discontinuities occurring due to the variation of shear

and internal forces in continuous beam are efficiently predicted using the developed

formulae. Museros et al. [62] have studied the free vibration analysis of a simply

supported structure under travelling mass.

Zarfam et al. [63] have explored the response spectrum of Euler-Bernoulli structure

subjected to time varying mass with the action of harmonic and earthquake support

vibration. Azimi et al. [64] have considered the effect of longitudinal acceleration of the

vehicle to formulate a numerical method for vehicle-bridge interaction dynamics. Cicirello

and Palmeriet [65] have studied the static vibration analysis of Euler- Bernoulli structure

with a random number of cracks under the action of both axial and transverse load. Zhong


12

et al. [66] have analysed the dynamic behaviour of the prestressed bridge and moving

vehicle using vehicle-bridge interaction dynamics. Costa et al. [67] developed a theoretical

formulation for the critical speed of traversing mass problem on elastic foundation. Fu

[68-69] has given the attention to the instant of switching of cracks and its effects on the

dynamic excitation of continuous bridge structure under travelling vehicles. In the later of

his investigation, he has formulated a numerical solution for a cracked simply supported

bridge with switching cracks under the excitation of seismic and the moving vehicle. Aied

et al. [70] have applied the ensemble empirical mode decomposition (EEMD) to study the

response of acceleration of a bridge structure subjected to moving load with the intention

of confining immediate change of bridge stiffness.

2.2.2 Finite element analysis/method for response analysis of structure

Using finite element method, Hino et al. [71] have determined the deflection and

acceleration of a reinforced concrete bridge structure under traversing vehicle load. The

analysis has been carried out with constant magnitude and speed considering the bridge

over River Bramphaputra in India. Bhashyam and Prathap [72] have presented the

efficiency of Galerkin finite element method to examine the large vibration amplitude of

fixed structures with different end conditions. Olsson [73] has formulated a method to find

the response of bridge with the action of moving load in modal coordinates. He has

established some of the finite element methods for traversing load problems. Yoshimura et

al. [74] have investigated the random excitation of the non-linear structure with different

types of sectional areas under a moving vehicle. The longitudinal and transverse

deflections of the beam are calculated using the Galerkin finite element method. Lin and

Trethewey [75] have presented a finite element formulation for structures under different

types of moving mass. Chang and Liu [76] carried out the random vibration analysis of a

nonlinear structure on an elastic foundation under traversing load. The deterministic and

statistical responses of the structure have been calculated by combining Galerkin method

and finite element analysis.

Rieker and Trethewey [77] have explored the finite element analysis of elastic structures

subjected to traversing distributed load. This work was modelled for the improvement of

the distributed railway train mass structure. Song et al. [78] have developed a novel finite

element model for three-dimensional finite element analyses to examine the response of a

structure subjected to high-speed trains. The objective of their work was to develop better

finite element models to be used in structural element railway bridge structure. The


13

developed finite element model was verified through numerical examples considering a

simply supported steel-concrete and a PC box-girder bridge structure. Law and Zhu [79]

have examined the response of cracked reinforced concrete structures under moving

vehicles in the presence of both open and breathing cracks. Experimental procedure has

been carried out on a Tee-section beam subjected to moving vehicles to ensure the

significance of crack models. The dynamic bridge deflection, change in relative and

absolute frequency, and phase plot of the dynamic responses of structures are studied for

the probable correlation of crack modelling as open or breathing crack.

Using the modal superposition and closed form solution method, Yang and Lin [80] have

determined the transverse deflection for a bridge and traversing vehicle through vehicle

bridge interaction dynamics. The deflection, speed and acceleration of the moving

structural system are calculated by the fundamental frequency of the bridge and the

driving frequency of the moving vehicle. Using the commercial software LS-DYNA, a

review of finite element analysis procedure was carried out by Kwasniewski et al. [81] to

study the vehicle-bridge interaction systems. The experimental test was conducted in a

bridge structure at Florida, US.

Ju and Lin [82] have developed a finite element model to explore the responses of vehicle-

bridge dynamics due to the application vehicle braking and acceleration. To verify the

finite element method for the effects of vehicle braking and acceleration, a two axle

vehicle traversing on bridge structure was formulated through semi-theoretical solution.

The application of this proposed method is limited to linear and small displacement

analysis. Bajer and Dyniewicz [83] have proposed an effective space-time finite element

method to solve the problems based on moving load. Kahya [84] has presented a multi-

layer shear deformable composite beam structure subjected to traversing mass using finite

element method. The consequences of moving speed and laminated lay-up on the response

of the beam structure with different end conditions have been determined. He has

concluded that angle-ply laminates and laminas stacking have a significant influence on

the beam response. Dyniewicz [85] has proposed the application of velocity formulation to

explain general moving inertial mass problem using space-time finite element method.

Esen [86] has developed a novel finite element method to examine the vertical vibration

of a rectangular plates structures excited by traversing load. Zhang et al. [87] have

formulated a finite element model for bridge structure using the equivalent orthotropic

material modelling (EOMM) for the application of multi-scale dynamic loading. The


14

dynamic responses and properties of a simplified bridge structure are attained using the

equivalent orthotropic material modelling method. Amiri et al. [88] have analyzed the

vibration based dynamic behaviour of a Mindlin elastic plate under the excitation of

traversing mass using the method of eigenfunction expansion. The first order shear

deformation hypothesis was applied to determine the response of plates subjected to

moving load excitation with different end conditions. Ju [89] investigated the

improvement of bridge structure for the safety of travelling trains during earthquakes

using finite element analysis. The interaction and separation of rails and wheels were

accounted in this analysis. Nejad et al. [90] have predicted the natural frequencies and

modal shapes of a double cracked beam in the theoretical formulation using Rayleigh's

method. This method has better applicability over eigenvalues method. But the accuracy

of this method is limited to small crack depth. Aied and Gonzalez [91] have explored the

response of a simply supported structure subjected to traversing load and determined the

deviation of strain rate and its consequence on the modulus of elasticity. The significance

of mass magnitude and speed on the dynamic deflection and strain rate of the structure are

also examined.

Wu et al. [92] have studied the dynamic behaviour concrete pavement structure subjected

to traversing load. The concrete pavement structure has an asphalt isolating layer on its

surface. The response of the structure has been carried out by finite element method using

the commercial ABACUS software. Stress and dynamic deflection of the concrete

structure are also determined at the critical moving load position by altering the depth,

modulus of elasticity of isolating layer and the amalgamation between the dividing layer

and concrete slab. Jorge et al. [93-94] have analysed the dynamic behaviour of structure

supported on nonlinear elastic foundation under travelling mass. The consequences of the

intensity of travelling load and speed and foundation’s stiffness on the response of the

structure have been investigated. The objectivity of this work is to study the structural

behaviour on any types of realistic foundation. Further, they have extended their research

work on the analysis of moving load structure problem on a bilinear foundation. The

critical speed of the moving load and deflection of the structure UIC-60 rail systems are

determined in finite element domain. Alebrahim et al. [95] have studied the dynamic

excitation of a self-hauling composite structure subjected to traversing load using transfer

matrix method and finite element method.


15

Ozturk et al. [96] have carried out the dynamic analysis of a hinged-hinged damaged

structure cracked on elastic foundation subjected to traversing load using finite element

method. The dynamic response of the structure has been calculated using Newmark-

integration method in finite element domain. The consequences of crack depth and

position, traversing mass speed, the elastic foundation on the beam dynamic deflection

have been verified. Fu [97] has examined the dynamic vibration of a simply supported

bridge structure with the existence of switching cracks under the action of traversing train

load and seismic vibration. The vibration of the bridge structure was studied by modal

analysis method in finite element domain. From the analysis of his work, he has remarked

that the switching crack can alter the stiffness of the structure due to seismic vibration, and

the deflection of the structure can amplify. Using finite element method, Beskou et al. [98]

have investigated the effects of 3-D pavement under traversing vehicle excitation. The

FEM was carried out in the time domain using the commercial ANSYS program for the

response of the vehicle-structure interaction analysis. They have observed that with the

increase in moving speed, the dynamic response of the pavement structure also increases.

Their work has been limited in linear elastic material properties.

2.3 Damage detection

2.3.1 Classical/FEA based methods for damage detection in structure

Dutta and Talukdar [99] have proposed a damage detection technique using the alteration

in dynamic response between the undamaged and damaged states. Eigen values analysis

has been carried out by employing the algorithm of Lanczos in adaptive h-version in finite

element domain to control the discretization fault to evaluate the modal parameters

precisely. Friswell and Penny [100] have compared various methods of crack modelling to

structural health monitoring problem using the low-frequency vibration and crack

flexibility models based on beam elements. The effects of breathing cracks on vibration

are also investigated on the bilinear stiffness of beam elements. Lee et al. [101] proposed a

damage identification method for bridge structures subjected to a passage of vehicle

loading using the data of ambient vibration. Abdo and Hori [102] have carried out

numerical formulation fault detection in a structure using the relationship between cracks

and dynamic properties changes. The damaged region has been found out using

characteristics of rotation of mode shape in this analysis. All these studies are carried out

in finite element analysis domain. A novel algorithm for crack detection and quantification


16

in the structure are formulated by Kim and Stubbs [103] using the alteration in modal

characteristics. Their algorithm was applied to detect the crack in two-span continuous

structure with the knowledge of pre and post-crack modal parameters. Chondros et al.

[104] have explored a continuously damaged beam vibration theory for the transverse

vibration of damaged beam with the presence of single and double edge open cracks.

Chondros and Dimarogonas [105] have studied the vibration analysis of cracked cantilever

beam with single edge open crack. The analysis has been further extended for the

detection of multiple cracks in the structure. Khoshnoudian and Eafandiari [106] have

developed a damage diagnosis method using the modal data and finite element analysis of

the structure. Swamidas et al. [107] have predicted the significance of crack size and

location of the fundamental frequencies of the structure using the Timoshenko and Euler

beam formulations theory. The consequences of cracks on shear deformation and rotary

inertia of the structure are also examined. Mazurek and DeWolf [108] conducted

experimental analysis to predict the possible damages in bridges using the vibration

signatures. Important efforts were given on automation, acquisition and study of vibration

signatures for up gradation of a bridge structure. Ruotolo and Surace [109] explored a

fault detection technique for a structure with multiple cracks using the modal data at the

inferior modes. Using the changes in flexibility, Pandey and Biswas [110] proposed a

damage identification method for structures and verified the method through experimental

tests. Using the Frobenius method, Chaudhari and Maiti [111] have examined the

vibrations analysis of a geometrically segmented slender structure in the transverse

direction with and without a crack. Considering the natural frequency as an input, they

have applied this method for crack detection and sizing. The problem identifying stiffness

changes in bridge structure under traversing oscillator have been proposed by Majumder

and Manohar [112] in time domain analysis. Chinchalkar [113] has developed a

computational method to determine the crack location in beam structure employing the

first three natural frequencies of the beam by finite element analysis.

Haritos and Owen [114] employed the vibration data for the assessment of damages in

bridge structures. The damaged identification is carried out using system identification and

statistical pattern recognition methods individually in a reinforced concrete bridges. They

have concluded that these two methods are to be complementary to each other and a good

approach for structural health monitoring problems related to bridges. A theoretical along

with experimental method has been employed for crack detection in the beam by Nahvi


17

and Jabbari [115]. This method was based on the measurement of natural frequency and

mode shapes. Alvandi and Cremona [116] have reviewed various structural fault detection

methods like flexibility change method, flexibility change curvature method, mode shape

curvature method, and strain energy method based on vibration signatures. From the above

fault detection methods, strain energy method is the effective one. Wang et al. [117] have

presented damaged detection method in structure named Local Damage Factor (LDF)

which can predict the existence, location and quantification of damage. For the

verification of the proposed method, a practical study was carried out in a 3-D steel frame

and wharf. Fushun et al. [118] have proposed a fault identification method in a bridge

subjected to a moving vehicle by introducing a method named ‘moving load damage-

locating indicator (MLDI)’. The values of modal curvature and MLDI at every node of the

baseline of the structure and the damage models have been calculated. The sudden

possible changes of MLDI values would give the possible location of the damage. Yu et

al. [119] have carried out a review analysis in the present information on factors

influencing the performance and identification of traversing loads in bridges.

Sekhar [120] has conducted various studies to examine the influences of cracks on a

structure and summarized different crack detection methods using vibration signatures.

Based on bending vibration measurement, a theoretical method for crack detection in

uniform structures with different boundary conditions with the presence of single edge

crack has been proposed by Khaji et al. [121]. The developed theoretical method has been

verified through numerical examples. Using the vibration amplitudes, Lee [122] developed

a method to identify multiple cracks in the beam by finite element method. The developed

method has been validated by Newton-Raphson method numerically. Zhu et al. [123] have

presented a novel technique to identify cracks in large-scale concrete columns for the

assessment of automated bridge conditions. The results from the developed technique have

been compared with manual detection of faults to check the precession of the method.

Zhang et al. [124] have presented virtual distortion method for instantaneous detection of

traversing mass and damages in structure from the dynamic response of the structure.

They have used the forces between the traversing mass and structure as excitation forces.

An analytical as well as experimental verification, have been conducted to identify the

crack in a simply supported beam using the natural frequencies of the cracked beam by

Sayyad and Kumar [125]. A relationship among location and size of crack and natural


18

frequencies of the beam has also been formulated. The analysis has been carried out in

finite element analysis and verified by experimentation.

Dilena and Morassi [126] have presented dynamic tests on a damaged bridge. The

deviations of modal properties at lower modes of vibration after enforcing the artificial

incremental damages are determined using the harmonic force tests. Roveri and Carcaterra

[127] have proposed an innovative method based on Hilbert–Huang transformation to

detect damages in bridge structure under traversing vehicle load. The forced response of

the structure has been found out by modal analysis method and the single-point response

obtained by the Hilbert–Huang transformation. The possible location of damage is

exposed by the assessment of the first instantaneous frequency curve which creates a

quick crest in the association of the damaged segment. Nguyen [128] has carried out a

comparison analysis for crack detection in structures under moving load with on open and

breathing cracks. The beam stiffness with an open crack has been determined from the

concept of fracture mechanics where as beam stiffness with a breathing crack has been

modelled as a time-dependent matrix based on the stiffness of the beam with open crack.

Li and Law [129] have formulated a substructural damage detection method subjected to

travelling vehicles using the dynamic response reconstruction procedure. The finite

element modelling of the undamaged structure and the determined dynamic acceleration

response from the damaged substructure are necessary for the damage identification in the

structure. Zhu and Law [130] have formulated a damage identification procedure of a

cracked simply supported structure under travelling load in the time domain using the

interaction forces between the travelling load and structure as the vibrating force. The

interaction forces between the vehicle-bridge and damage of the structure on the bridge

deck are recognized from the calculated responses of the damaged bridge through

succession iterations without the information of the moving loads.

Khiem and Lien [131] investigated multi-cracked identification problem for one-

dimensional structure using natural frequencies. Patil and Maiti [132] have developed a

crack detection method to predict the position and quantification of cracks in a slender

cantilever beam. The method has been formulated using the natural frequencies of the

beam and later verified through experimental tests. Pakrashi et al. [133] developed an

experimental investigation of cracks in bridges using the vehicle-bridge interaction forces.

This analysis could be helpful for the online health monitoring problem of bridges under

moving vehicle. Zhou et al. [134] have developed a magnetic wire smart film for crack


19

detection in large-scale concrete bridges. The smart film inventions, the basic operational

principle of the film, and the film application on an actual bridge structure have been

explained. Bouboulas and Anifantis [135] have developed a finite element model to

analyze the behaviour of a vibrating beam with the presence of non-propagating edge

crack. The response of the beam is studied using either Fourier or wavelet transformation

to evaluate the effects of breathing cracks. The consequences of the angle of cracks, crack

depth and position, are investigated on the vibrational behaviour of the cracked structure

through various parametric studies. Zhan et al. [136] have proposed a fault isolation

method for railway bridges by utilizing the train-induced dynamic responses and

sensitivity investigation. The significance of measurement of noise and track irregularities

is explained in the proposed method.

Ghadami et al. [137] have presented an easy method to identify, locate and quantify

multiple cracks in the structure using the measured natural frequencies. The key

objectivity of the proposed method is to identify the unknowns of cracks intervention. Hu

and Liang [138] have proposed an integration method based on the theory of vibration to

detect arbitrary number of cracks in structure. The utilization of massless insignificant

springs and the continuum of damage conceptions have been integrated to locate the

potential damages. Chomette et al. [139] have investigated an innovative method to

identify tiny cracks on structure using active modal damping and piezoelectric mechanism.

The difference in active damping is identified using the Rational Fraction Polynomial

method as a pointer of cracks detection in the proposed method. Nejad et al. [140] have

approximated a theoretical method by extending Rayleigh's method for a structure with

single or double cracks to determine the natural frequencies and mode shapes of that

structure. Wang et al. [141] have developed a fault diagnosis system in beam type

structure using the statistical moment analysis. Numerical analysis has been carried out on

a damaged simple supported and two-span continuous structure to show the accuracy of

the formulated method. Using the finite element method, Nguyen [142] has analyzed the

mode shapes of a damaged beam and applied it for damage identification. The significance

of the coupling system between longitudinal and transverse bending vibrations due to the

presence of crack on the mode shapes has been inspected. This quantitative mechanism

has also been employed to estimate the size and location of the crack with the pragmatic

coupled modes.


20

Lee and Wu [143] proposed a simplified method to replicate multi-cracks in structure.

Local adaptive differential quadrature method has been developed to find out the natural

frequencies of the beam. The position and depth of cracks have been predicted using the

Newton–Raphson iteration analysis. Two crack transfer matrix has been investigated in

finite element method for prediction of cracks in a structure by Nandakumar and Shankar

[144]. The developed damage identification procedure was also verified through

experimental tests using the local detection of sub-structure of a fixed beam. This method

is also applicable for damage detection in large structures. Khiem and Tran [145] have

investigated a crack isolation method for structure with arbitrary number of cracks using

vibration mode. The crack detection and localization have been done effectively with this

method from the sparsely and noisy data also. Ettefagh et al. [146] have designed an

innovative damage identification algorithm based on method of model updating. The

design method has been employed to replicate the deteriorated structure considering

various levels of noise to check the efficiency of this method. Schallhorn and Rahmatalla

[147] have proposed a vibration based damaged identification procedure to access the

health monitoring condition of a highway steel-girder bridge.

He and Zhu [148] have explored a closed form solution to analyse the response of a

deteriorated simply supported structure under traversing mass and investigated the

consequences of local stiffness loss based on the traversing frequency element

corresponding to the traversing load and the natural-frequency element of the beam. A

deep insights into structural damage identification based on the traversing mass-induced

dynamic response has been developed using this method. The consequences of altering

traversing speed and traversing load dynamics on damage localization are also explained.

Zhu et al. [149] have determined the local damage of the large and intricate structure

subjected to traversing load. Law and Lu [150] have proposed a time domain analysis to

identify crack parameters from the measurement of beam deflection. The response of the

moving load structure is calculated by modal superposition method. The presented damage

detection method has been proved with experimental works through impact hammer test

on a structure with single crack. Using alteration in the nonlinear characteristics structures,

Law and Zhu [151] have developed a damage detection procedure of a reinforced concrete

beam structure subjected to moving load. Nonlinearities characteristics of the beam

structure have been identified by considering the alteration in the instantaneous frequency

at different positions of the vehicle load along the beam. Bu et al. [152] have proposed a


21

novel approach to assess the damage occurring in a bridge deck using the measured

dynamic response of a moving vehicle on a structure. The vehicular load has been treated

as a smart sensor and force transducer in the moving load interaction structure to detect

the damages.

Karthikeyan et al. [153] have developed a method to know the location and quantification

of cracks in a structure using the modal parameters in finite element model. A vibration

based damaged detection method has been recommended by Zhou et al. [154] to estimate

the position and quantification of damages on a bridge deck. The analysis has been carried

out in finite element analysis as well as laboratory test experimentation. Based on

mathematical modelling, Al-said [155] has intended a crack detection method to locate

and quantify cracks in a stepped beam under a slowly traversing mass. In the proposed

method, the crack is identified by monitoring a single natural frequency of the system. The

validation of the method has been done with experimental test along with finite element

analysis. Yin and Tang [156] have carried out a computational study to detect multiple

cracks in a cable-stayed bridge using the transverse deflection of a vehicle crossing over a

bridge. Li et al. [157] have performed a numerical analysis along with experimental

verification to estimate damages in a structure under a moving load. Experimental analysis

has been conducted on a Tee-section concrete structure subject to a moving vehicle for the

accuracy of the proposed method. Li and Hao [158] have used the relative displacement

measured from the vibrant response of the passing vehicle on the structure as damage

indicator. Using the vehicle transmissibility of vehicle-bridge coupled structure; Kong et

al. [159] have studied a fault diagnosis method for bridge structure under a passing

vehicle. Various types of damage indicators are prepared artificially and based on the

transmissibility of the response of the vehicle and bridge structure, and a damage detection

procedure has been conducted numerically.

Aydin [160] has recommended an effective procedure to determine the vibration

characteristics of a flexible structure subjected to axial load. Dar et al. [161] have

performed an experimental study on the dynamic behaviour of a deteriorated bridge

structure. The objectivity of the work was to find the load-carrying capacity of the bridge.

Oshima et al. [162] have proposed a damage assessment method for bridge structure using

the estimated mode shapes due to the traversing vehicle. The proposed method has been

proved by various numerical solutions. Mazanoglu [163] has recommended crack

detection procedure for multiple cracks based on the concept of measurement of natural


22

frequency ratios. The procedure was verified through frequency ratios obtained from the

experimental test. Feng and Feng [164] have proposed a damage isolation procedure for

bridge structure by utilizing the vehicle induced deflection without the knowledge of the

traffic-induced vibration and surface roughness. The probability and performance of the

developed method have been validated by numerical simulation with single and multiple

cracks.

2.3.2 AI techniques based methods for fault detection in structure

2.3.2.1 Genetic Algorithm based methods for crack detection in structure

Chou and Ghaboussi [165] have formulated an optimization problem using the genetic

algorithm (GA) for damage detection in structure. The deflections at unmeasured degrees

of freedom have been calculated by GA to avoid the analysis of structure in fitness

estimation. Shopova et al. [166] have solved various types of optimization problem using

GA. A common genetic algorithm problem has introduced the basic principle of the

number of selection, reproduction and mutation parameters in the correspondence, genetic

operators. He et al. [167] have recommended a bridge damage detection technique using

the vehicle induced bridge response as a direct problem and genetic algorithm as a reverse

problem. The genetic algorithm is employed to identify the pattern and location of

damage. Sun et al. [168] have designed a vehicle suspension system using the genetic

algorithm and chosen the minimum dynamic pavement moving load as the design

principle. Baghmisheh et al. [169] have proposed a fault isolation method for cracked

structure based on genetic algorithm. A Genetic algorithm has been used to examine the

natural frequencies of the cracked structure. The position and depth of cracks in the

structure are identified by the continuous genetic algorithm optimization method. Meruane

and Heylen [170] have developed a hybrid coding based on genetic algorithm to identify

the cracks parameters in structures. The proposed method can also identify the happening

of false damage due to the errors in the noisy experiment and numerical simulations.

Buezas et al. [171] have employed a finite element method along with a formulated

genetic algorithm for fault diagnosis in damaged structure. The formulated optimization

method can identify not only beam like structure but also structure like curved beam and

blade like structural element with rotational motion.

Na et al. [172] have investigated a damage detection procedure for a shear building using

genetic algorithm followed by the flexibility matrix with the dynamic response. The


23

proposed method has been exemplified with numerical analysis. Mehrjoo et al. [173] have

applied the genetic algorithm as inverse method to estimate the location and severity of

cracks in cracked Euler-Bernoulli beam. The inverse method has been verified with

different types of damage scenarios. Lee [174] has applied the finite element method as

direct method and coupled genetic algorithm as the reverse method to detect the traversing

mass on the bridge deck. This method can also identify the mass of the moving load as

well as the traversing speed. Chisari et al. [175] have developed a fault isolation method

for a base-isolated and post-tensioned concrete bridge using the genetic algorithm. The

algorithm was based on static and dynamic loading condition.

2.3.2.2 Neural network based methods for crack detection in structure

Alli et al. [176] developed a method based on artificial neural network (ANN) to solve the

dynamic systems problems. The developed method has been applied for the solution of

vibration control problem. Cao et al. [177] have recommended a method to detect the

flight loads on aircraft wings based on ANN approach using the relationship of load-strain

analysis in structures. Mahmoud and Kiefa [178] have explained vibration problem on

cracked structure employing general regression neural networks (GRNN). The accuracy of

the employed method has been checked with a cracked cantilever beam with an edge

crack. The location and size of the crack have also been identified. Waszczyszyn and

Ziemianski [179] have presented some fundamental concepts on back-propagation neural

network (BPNN) to study the bending effects on elastoplastic structure, problems on plain

stress, the fundamental concept on vibration analysis of real building and damage

identification. Zang and Imregun [180] have jointly approached the ANN and principal

component analysis (PCA) for structural damage detection by reducing the frequency

response functions (FRFs). The FRFs have been used as the input parameters to the

network and damage parameters as output.

Liu et al. [181] have employed a BPNN and computational mechanics to identify cracks in

the structure. The types of cracks present, the extent of cracks, location and severity of

cracks are evaluated using the approached BPPN. Chen et al. [182] have developed a fault

isolation method of structure based on neural network using the response data and

transmissibility function as input to train the proposed network. Kao and Hung [183] have

investigated a neural network based damage detection procedure using the changes in

properties of unknown structural systems. This procedure involves two systems namely

system identification which detects the uncracked and cracked states of structural systems


24

and structural damage detection. Sahin and Shenoi [184] have presented a method to

quantify and localise the damages in a beam like structure using ANN and verified it

through experimental validation. The alteration in first three natural frequencies and

curvature mode shapes of the structure obtained from FEM have treated as the input

parameters for the proposed ANN and in the later part, the location and quantification of

the cracks have been predicted by the proposed ANN.

Ataei et al. [185] have applied the linear two layer feed forward neural network (FFNN)

with back propagation learning technique to measure the stain and deflection from railway

bridge load test. Kang et al. [186] have proposed a BPNN to estimate the fatigue life of a

structure subjected to multi-axial loading. The prediction of the fatigue life period was

based on critical plane concept using finite element modelling. Yan et al. [187] have

summarized various methods for structural damage detection based on vibration

signatures. It covers the theory of intelligent damage isolation method and its application

in prediction in structural damage recognition. Li and Yang [188] explored an innovative

technique for structural damage diagnosis using ANN with arithmetical properties of

structural dynamic induced responses. The alterations of variances or co variances of

responses of structure are chosen as damage parameters for damage isolation. The BPNN

with the alteration of variance of responses of structure as input parameters and damage

index as output parameters are applied for damage isolation in structure. Mehrjoo et al.

[189] investigated a method for prediction of the crack intensities of joints of truss bridge

structure by applying BPNN. The exactness and effectiveness of the adopted BPNN

method was exemplified with numerical analysis. Bakhary et al. [190] approached a

method to estimate petite structural damage based on ANN in association with multi-stage

sub-structuring. The position and extent of the damages are identified by substructure

method with multi-stage ANN modelling. The approached method was experimentally

verified by considering continuous concrete slab and a three-storey portal frame structure.

Al-Rahmani et al. [191] have investigated a combined approach for fault detection in

bridge structure using soft computing mechanism. The mechanism was based on

ABAQUS finite element analysis and ANN methods. Using the arithmetical properties

obtained from the dynamic response of a moving train, Shu et al. [192] have developed

ANN-based method to diagnose structural defects in a simplified railway bridge structure.

The analysis has been carried out for a single span bridge. Yaghinin et al. [193] have

proposed a hybrid algorithm based on the application of the training of the ANN and


25

explained the exactness, different mechanism and validation of the developed algorithm.

Elshafey et al. [193] have estimated the crack spacing in a concrete structure using neural

network. Erkaya [194] has predicted the vibration characteristics of a gear system using

neural networks. The proposed network has been trained with Levenberg–Marquardt

learning mechanism. Pandey and Barai [195] have developed the method of multilayer

perceptron learning to identify multi-damages in bridges. The developed method was

applied to a truss bridge structure. Karninsk [196] proposed a method using ANN as a

predictor to localize the damage employing the change of natural frequencies as inputs.

This method has been applied to a free-free beam for the exactness of the ANN model.

Seibi and Al-Alawi [197] estimated the toughness of fracture using ANN and analyzed the

consequences of crack geometry, temperature and toughness of fractures. Zhao et al. [198]

have proposed a computational intelligence based NN method to diagnose faults in

various types of structures like a beam, frame and support movement of beams. The

requisite data to train the ANN model were obtained from FEA. Marwala and Hunt [200]

have employed the frequency responses and modal parameters obtained from FEA and

ANN to estimate faults in damage vibrating structure. Chang et al. [201] have proposed an

iterative ANN model for structural damage identification. An adapted back- propagation

mechanism has been developed to estimate the damage parameters and verified with the

experimental investigation. Lin et al. [202] have investigated an NN- based method to

estimate damages of bridges during major earthquakes. The proposed method was applied

to bridges in Taiwan to predict the seismic damages. Zang et al. [203] approached

combined efforts of independent component analysis and ANN for structural fault

diagnosis. The exactness of the proposed method was verified in real life study

considering truss and bookshelf structures. Yeung and Smith [204] investigated a crack

detection procedure using neural network by employing vibration signatures obtained

from FEA. Using the errors occurred in baseline FE model, Lee et al. [205] explored fault

diagnosis procedure for the bridge under passing vehicle. The developed method has been

verified in a laboratory test considering a simply supported and multi-girder bridge

structure. Bakhary et al. [206] have proposed an NN model based on statistical properties

to detect damages in structure considering the uncertainties occurred in the NN model.

The proposed approach was verified experimentally with a steel portal frame and concrete

slab model.


26

Wong et al. [207] developed an algorithm based ANN for online identification of damages

to structure induced due to ground shaking. A 2-story steel-frame building was used to

check the accuracy of the approached algorithm. Pawar et al. [208] applied spatial Fourier

analysis along with NN to analyse faults in structures. The approaching methods were

verified through numerical examples with the presence of noises. Gonzalez-Perez and

Valdes-Gonzalez [209] presented ANN based structural damage detection technique to

bending in the girders of bridge subjected to vehicle loading. The differences in modal

strain energies are considered as inputs to the ANN model with 12,800 damages scenarios

in the proposed technique. Li [210] proposed a probabilistic neural network approach for

localization of defects in the composite plane structure. Parhi and Dash [211] have carried

out finite element modelling along with neural network based damage prediction to

identify multiple cracks in the beam-like structures using the vibration signatures as inputs

to the proposed ANN model. The feed-forward multi-layered neural network based on

back-propagation mechanism was applied for the estimation of crack parameters. Elshafey

et al. [212] have estimated the crack width in a concrete structure by ANN. The radial

basis and feed-forward neural networks with back propagation algorithm were applied to

predict the crack width in the structure. Hasancebi and Dumlupınar [213] have applied the

nonlinear finite element modelling along with ANN to analyze the load rating of bridge

structures. The 3D FE model along with experimental verifications has been carried out to

determine the nonlinear response and load rating of single span T-beam Bridge.

Using frequency response function, Bandara et al. [214] have applied ANN method to

identify defects in a structure for a given label of vibration. The key objective of the

proposed work is to analyse a feasible technique for structural health monitoring based on

vibration responses which would reduce the elements of the initial frequency response

functions. Aydin and Kisi [215] have applied the Multi-layer perceptron and radial basis

neural networks based mechanism for fault isolation in the beam-like structures. The

modal properties of the structures as input parameters have been provided to the proposed

networks. It has been shown that the radial basis neural networks have performed better

and can be used as damage identification algorithm. Kourehli [216] has approached an

innovative idea for fault detection in structures using the incomplete modal data and ANN.

The ANN model has been trained by the first two incomplete mode shapes, and natural

frequencies of the structure obtained from the finite element model with feed forward

back-propagation mechanism. The accuracy of the model was validated with a three-story


27

plane, spring-mass and simply supported structures. Alavi et al. [217] have integrated the

finite element method along with probabilistic neural networks with the concept of

Bayesian decision mechanism to identify damages in structures. The exactness of the

innovative technique was primarily estimated in a simply supported structure under three-

point bending and later in a bridge gusset plate.

2.3.2.3 Recurrent neural network based methods for crack detection in structure

Yu et al. [218] have applied Elman’s recurrent neural network (RNN) to estimate the

performance of a boring mechanism through its full life cycle. Xiong and Withers [219]

have proposed an RNN model to predict the progression of the damages produced during

hot non-uniform and non-isothermal forging processes. The hyper parameters related to

the noise level and weight decay of the proposed RNN model has been trained with

introducing the Bayesian algorithm. Ekici et al. [220] developed an Elman’s RNN based

method to estimate the locations of transmission line fault. Wavelet transformation

method was implemented to select characteristic description about the faulty signals. The

developed ERNN model can predict the fault locations quickly and an alternative

characteristics to the feed forward back propagation networks and radial basis functions

neural networks. Abdelhameed and Darabi [221] have applied the RNN based mechanism

to control the fault-tolerant of mechanized in order manufacturing systems under sensor

faults. The training procedure of the applied RNN model has been performed based on

training data produced from the well-mechanized system run by a programmable logic

controller. Connor et al. [222] have developed a healthy training mechanism and applied it

to RNNs. The proposed mechanism was based on filtering outliers from the induced data

and predicted the required parameters from the filtered data. Pearlmutter [223] has

determined the gradient of a dynamic RNN using back propagation algorithm. Tse and

Atherton [224] have predicted the deteriorated conditions of machine structures using

vibration parameters and RNN. The significance of defects and remaining life span of

machine structures are also estimated applying the proposed method.

Gan and Danai [225] have proposed a rule based RNN to model dynamic systems. The

rule based RNN model has been designed on linearized state space modelling of the

dynamical system. Waszczyszyn and Ziemianski [226] have applied various types of

mechanics to ANN and RNN model to analyze faults in different types of machine

structures using back propagation algorithm. Valoor et al. [227] have developed a self-

adaptive vibration control structure using RNN to control the excitation of beam


28

structures. Seker et al. [228] have used ERNN for the health monitoring of nuclear power

plant and rotating machines. Schafer and Zimmermann [229] have proved the RNN as

common approximators and showed its capacity in state-space model. Thammano and

Ruxpakawong [230] have introduced the concept of additional weight to the standard

JRNNs and ERNNs. Coban [231] has proposed a rule based context layer locally RNN to

identify faults in dynamic systems using back propagation mechanism. The accuracy of

the proposed model has been verified with the experimental application with a D.C motor.

2.4 Statistics based method for fault identification in

structure

Sohn et al. [232] have employed the statistical process control approach to structural

health monitoring problem. The approach was exemplified by replicating online

monitoring of damages in structures. Fugate et al. [233] have proposed a statistically based

damage isolation method using vibration signatures. The measured data are obtained from

the acceleration-time histories of undamaged structure based on the autoregressive model.

Residual errors which differentiate between the predictions from the autoregressive model

and the real data measured from the time- history are implemented as damage responsive

characteristics to the structures. Lei et al. [234] have developed a statistics based fault

diagnosis analysis using the time series prediction method on vibration signatures. Lu and

Gao [235] proposed an innovative time-series model to diagnose damages in structures.

The proposed model was originated in an exogenous form containing the acceleration

responses.

Mattson and Pandit [236] developed a mechanism based on vector autoregressive (ARV)

model to identify damage locations in vibrating structures. The existence and location of

damages are indicated by the standard deviation obtained from the residual data of the

ARV. Zhang et al. [237] proposed an inventive damage isolation mechanism using the

statistical moments of dynamic analysis of a structure subjected to arbitrary excitations.

The sensitivity of the proposed method to structural damage has been examined for

different types of structural responses and various orders of the statistical moment. The

method has also been extended from SDOF to MDOF systems. Gul and Catbas [238] have

carried out theoretical analysis followed by experimental verification for structural health

monitoring problems using time series modelling based on Statistical pattern recognition

method. The data obtained from ambient vibration are used to explain the proposed


29

methodology. Law and Li [239] have assessed the condition monitoring of concrete bridge

structure by estimating the reliabilities of the structure. The mean and the standard

deviations values of the estimation results of the structure are determined and then

incorporated in the reliability analysis of the structure. The approach has been verified

with FEA considering a concrete bridge under the action of a moving vehicle.

Zapico-Valle et al. [240] have developed an inventive damage isolation procedure for a

cracked cantilever beam based on statistical process control-based method and verified

with experimental analysis. The concept of signal length has been established as the

characteristic for statistical process control. Kwon and Frangopol [241] have predicted the

life period performance estimation and management of aging steel bridge structures under

damages by combining reliability model, crack growth model and probability of detection

model as prediction models. Mosavi et al. [242] have developed an autoregressive model

to identify the crack location in structures using the vibration response data. Cavadas et al.

[243] have applied the data-driven method on traversing-mass responses to identify

damage existence and location in structure. The proposed data-driven method consists of

two components namely moving principal component and robust regression analysis for

the condition monitoring of the structure. A novel statistical time series mechanism with

the functional model have been developed by Kopsaftopoulos and Fassois [244] to

identify, locate and estimate damages based on vibration induced response of the structure.

Phares et al. [245] have conducted field verification on bridge damage isolation relied on

statistically based algorithm. The method can precisely identify the damage of the

structure using the strain data accumulated from various sensors on the bridge. Wang et al.

[246] have developed a two step method using the concept of statistical moment for crack

identification in beam type structure. Yu and Zhu [247] have applied the time series

analysis and the higher statistical moments based structural responses to identify nonlinear

damages in structures. Reiff et al. [248] have developed a statistical bridge damage

detection procedure employing the girder allocation factors under moving load excitations.

This work has been designed to alert the conditions of bridges and present a better

probabilistic approach to assess the damage conditions and localizations.

2.5 Miscellaneous methods

Masciotta et al. [249] have approached a fault isolation method using the second order

spectral characteristics of the nodal response of structures. The precise dependences on the


30

frequency content of the outputs of power spectral densities are used as suitable

parameters for the identification and localization of damages. The method has been

validated through computational simulation considering the Z24 Bridge in Switzerland.

Dilena et al. [250] applied the interpolation damage isolation method for health

monitoring of structures based on frequency response function. The analysis has been

carried out for a single span bridge with growing levels of intense damages. Gokdag [251]

has investigated a crack detection procedure for structures subjected to moving vehicle

employing the particle swarm optimization method. This method can identify cracks up to

the relative crack depth of 0.1 even if in the presence of noise interference. Kang et al.

[252] have developed an improved particle swarm optimization procedure for crack

identification in beam type structure using vibration signatures. The improved method has

formulated combining the particle optimization method with the artificial immune system.

Hester and Gonzalez [253] have proposed an inventive wavelet analysis for damage

detection in bridge structure under vehicle load. The proposed method has used the

vehicle-bridge finite element interaction structure, acceleration signal and wavelet energy

content at each segment of the bridge for the structural damage isolation.

Sahoo and Maity [254] have developed a hybrid neuro-genetic mechanism for damage

isolation in different types of structure. The frequency and strain as input factors and the

damage location and severity as output factors are considered in the proposed network.

The accuracy of the network has been proved by exemplifying a fixed-free and plane

frame structure. Li et al. [255] have proposed a crack extraction method based on image

processing by placing a long distance acquisition device. The width of cracks is also

measured by implementing image clip & fill and rotation of transformation. Adhikari et

al. [256] have developed an improved model based on digital image processing for bridge

assessment. A 3-D visualization method has been formulated in such a way that the crack

can be seen as in the actual onsite visualization. Li and Hao [257] have investigated the

condition monitoring of truss bridge structure using relative displacement sensors. The

sensitivity and performance at the joint of the truss bridge structure are examined using the

displacement of ambient vibration.

2.6 Summary

From the literature review, some methods including numerical, finite element methods and

experimental analysis have been explored to study the response of the damaged structure


31

subjected to moving mass. However, according to the author’s knowledge, the numerical

procedure, FEA and experimental verifications have not been applied simultaneously to

determine the response of deteriorated structures under moving mass with multiple cracks.

Again there are so many techniques based on FEM, experimental investigations,

computational analysis, AI- based techniques, and statistics based methods are applied for

fault isolation in deteriorated structure subjected to moving mass. But so far from the

literature review, the application of rule-based Recurrent Neural Network (Jordan and

Elman), hybridisation of rule based Jordan’s and Elman’s RNN, a rule-based statistical

properties based method and hybridisation of rule- based RNN and statistical properties

based methods are scanty for fault isolation in deteriorated structures subjected to moving

mass.

32

Chapter 3

THEORETICAL-NUMERICAL

ANALYSIS OF MULTI-CRACKED

STRUCTURES SUBJECTED TO

MOVING MASS

3.1 Introduction

The dynamic interaction between a structure and the moving mass characterizes a

particular topic in the study of structural dynamics. Theoretically, the two subsystems

(structure and moving mass), can be replicated as two elastic substructures with a general

interface. Each substructure is characterised by some frequencies of vibration. The two

substructures act together with each other through the common contact forces. In this

Chapter, a theoretical-numerical investigation of structures with different end conditions

subjected to moving mass is presented.

3.2 The Problem Description:

Here, a computational solution to the system of partial differential equation has been

developed illustrating a multi-cracked structure subjected to a traversing mass at various

end states. The computational method expresses the conversion of the well-known

principal partial differential equations of motion into a novel solution of ordinary

differential equation. A cracked cantilever structure subjected to a traversing mass is

explained in Figure 3.1. Structures with different boundary conditions are considered in

the present dissertation. The basic objectives of the problem are as follows-

(i) Formation of easy and realistic theoretical-numerical method for evaluating the

responses of multi-cracked structures subjected to transit mass with different end states.

(ii) Influences of moving mass, speed, crack depth, and crack locations and their effects on

the solution of the problem.

Chapter 3 Theoretical-Numerical Analysis of Multi-Cracked Structures

Subjected to Moving Mass

33

3.3 The Problem Formulation

Structures with different end conditions are analysed by considering the fundamental

assumptions of the Euler-Bernoulli’s beam theory. The material is assumed homogeneous

and isotropic. The rotary inertia, shearing forces, damping, and longitudinal vibrations of

the beam structure are neglected, and the transverse vibration of the beam is considered.

The inertial and shearing effects of the moving mass are accounted in the current analysis.

From the analysis of Figure 3.1, the following assumptions are made:

( )u x u = Deflection due to longitudinal vibration of the beam.

( )y x y = Deflection due to transverse vibration of the beam.

( )I x I = Moment of inertia of the beam.

( )m x m = Beam mass per unit span length (constant)

g = gravitational constant, L =Length of the beam.

B=Width of the beam, H=Thickness of the beam.

( )F t The interactive force involving the moving mass and the structure.

1 2 3 1,2,3, ,L L L L = Position of the first, second and third cracks at the left end of the

cracked structures respectively. 1 2 3 1,2,3, ,d d d d = Depth of the first, second and third

cracks respectively. M = Mass of the moving mass, v =Moving speed of the transit mass.

vt = Position of the moving mass at any instant time‘t’.

h = The point of attention where the beam deflection is to be determined.

Based on the above assumptions, the governing equation of motion of a structure under a

moving mass is given as- 4 2

4 2( , )

y yEI m F x t

x t

(3.1)

Here ( , ) ( ) ( ) ( , )F x t P t x r x t

( , )r x t Standard loading conditions (zero in the present analysis). δ=Dirac delta

function.

3.4 Analysis of Cracked Structures Subjected to Moving

Mass

The present section is focused on determining the responses of multi-cracked structures

with different end conditions subjected to a moving mass. Initially, the analysis is carried

out for a multi-cracked cantilever beam under moving mass and it is further extended to



34

multi-cracked simply supported beam and fixed-fixed beam under a moving mass. The

mass ‘ M ’ is moving with a speed ‘ v ’ from the fixed end of the structure to the free end as

shown in Figure 3.1. The cracks with arbitrary crack depth are located from the clamped

end of the beam.

Based on the above postulations, the equation of motion of the structure under a transit

mass is expressed as 4 2

4 2( ) ( )

y yEI m P t x

x t

(3.2)

Here

2

( , )( ) v y tt

P t Mg M

Employing the integral properties of function of Dirac delta to the structure, it may be

expressed as-

Figure 3.1: Multi- cracked cantilever beam under moving mass

M

1d

2d

3d

v

1L

2L

3L

L

u

y

( )F t

( ), ( )I x m x

x

h

H

1,2,3d

B



35

0 0 0

0

( ) 0 ( ) ( )

L

f x xf x x x dx L (3.3a)

0 0 0

0

( ) ( ) 0, 0,

L

x xf x x x dx L (3.3b)

Substituting the value of ( )P t in equation (3.2), one can express as-

2

2

24

4

( , ) ( )

( , )( , )

y x tEI m Mg M x

t

y x tv y t

x t

(3.4)

The universal elucidation of equation (3.4) may be written as in series from i.e.

1

( , ) ( ) ( )n n

n

y x t x Q t

(3.5)

Where ( )n x = Eigen function of the beam without considering the moving mass.

( )nQ t = Amplitude function to be evaluated, n = Number modes of vibrations.

To evaluate ( )x , the equation (3.5) can be expressed as-

4( ) ( ) 0 iv

n n nx x (3.6)

Where 2 2

4 n n

nA

EI EIm

Due to the occurrence of three numbers of cracks in the structure, the complete structure

can be replicated by the combination of four structural segments, and each segment is

obeying the adopted assumptions. In view of the beam theory of Euler-Bernoulli, the

universal solution of equation (3.6) for determining the transverse deflections for each

segment of the structure may be expressed as:

1 1 1 1 1( ) sin( ) cos( ) sinh( ) cosh( ) , 0 n n n n n

x A x B x C x D x x L (3-7a)

2 2 2 2 1 2( ) sin( ) cos( ) sinh( ) cosh( ) , n n n n nx A x B x C x D x L x L (3-7b)

3 3 3 3 2 3( ) sin( ) cos( ) sinh( ) cosh ( ) , n n n n nx A x B x C x D x L x L (3-7c)

4 4 4 4 3 ( ) sin ( ) cos( ) sinh( ) cosh ( ) , n n n n nx A x B x C x D x L x L (3-7d)

Where , , &A B C D are constant coefficients of integration, computed from different

boundary conditions of the structures [Thomson, 258]. The modelling of cracks are

explained in Appendix-A. The natures of the cracks are open and transverse.

The different boundary conditions of the cantilever beam are as follows-

At 0, (0, ) 0, (0, ) 0x y t y t and at , ( , ) 0x L L t , ϐ(L, t) =0

Where ϐ is the sheer force and is the bending moment.



36

Substituting the equation (3.5) on the right of the equation (3.4) and arranging it, the

equation now can be written as-

2

1 1

( ) ( ) ( ) ( ) ( )n n n n

n n

Mg M v Q t x x tt

(3.8)

With multiplication of ( )P x in the equation (3.8) and integration of the equation upon the

entire length of the beam, the equation now can be expressed as:

2

10 0 0

( )( ) ( ) ( ) ( ) ( ) ( ) ( )

L L L

p p n n p n n

n

Mg M x dxx dx x v Q t x x t dxt

(3.9)

The term on the right of the equation (3.9) can be formulated as-

1 0

( ) ( ) ( )

L

n p n

n

dxt x x

(3.10)

From the orthogonality theory and orthogonal properties, the above term can be expressed

as-

1 1 2 2

0 0 0

0

( ) ( ) ( ) ( ) ( ) ( ) ... ( ) ( ) ( ) ...

= ( ) ( ) ( ) (Vanishing the lower terms)= ( ) ( )

L L L

p p p p n

L

p p n p p p p

t x x dx t x x dx t x x dx

t x x dx t S S t

(3.11)

It is due to the reason that 0

0, ( ) ( )

,

L

p n

p

n px x dx

S n p

Recalling the theory of orthogonality, orthogonal properties and Dirac-delta function’s

integral properties, and the equation (3.9) may be expressed as-

2

1

( ) ( ) ( ) ( ) ( )p n n p p p

n

Mg M v Q t t St

(3.12)

2

1

( ) ( ) ( ) ( )p n n p

np

Mt g v Q t

S t

(3.13)

The equations (3.4) and (3.8) may be united as-

4 2

4 21

( , ) ( , )( ) ( )n n

n

EIy x t y x t

m x tx t

(3.14)

Uniting equation (3.13) and (3.14) with ‘q’ number of stages, one can express now-



37

24 2

4 21 1

( , ) ( , )( ) ( ) ( ) ( )n q q n

n qn

my x t y x t M

EI x g v Q tx t S t

(3.15)

Substituting equation (3.5) in (3.15), the equation may be written now-

4 2

1 1

4 2

2

1 1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

n n n n

n n

n q q n

n qn

EI

x Q t x Q t

mx t

Mx g v Q t

S t

(3.16)

Substituting the values from the equation (3.6) in (3.16) and rearranging the equation-

2

4

,

1 1

( ) ( ) ( ) ( ) ( ) ( ) 0n n n n tt q q n

n qn

Mx EI Q t mQ t g v Q t

S t

(3.17)

The equation (3.17) have to satisfy for each values of ‘ x ’ i.e.

2

4

,

1

( ) ( ) ( ) ( ) 0n n n tt q q n

qn

EIM

Q t mQ t g v Q tS t

(3.18)

The equation (3.18) is valid to evaluate the responses of any types of structures subjected

to moving mass based on the above assumptions. The value of ‘ ( )nQ t ’ has been found out

by solving equation (3.18). Runge-Kutta fourth order technique has been employed to

explain the above equation in MATLAB domain.

3.5 Numerical Formulation of Cracked Cantilever Beam

under a Moving Mass

Numerical examples are formulated for the analysis of the proposed theory. A mild steel

cracked cantilever structure under a traversing mass has been considered with the

following dimensions.

L=100cm, B=3.9cm, H=0.5cm. M =1 kg and 2 kg, 1,2,31,2,3

dH

= Relative crack

depth=0.6, 0.25, 0.45 and 0.3, 0.55, 0.4. v = 438cm/s and 573 cm/s. Relative crack

location. 1,2,31,2,3

L

L =Relative crack Positions of first, second and third cracks from the

fixed end respectively=0.25, 0.45, 0.65 and 0.5, 0.65, 0.85.

The deflections of the cracked cantilever beam are calculated at different positions of the

moving mass along with the free end and shown in the following Figures for analysis.



38

Figure 3.2: Graph for Deflection of beam vs. Travelling time for

undamaged beam for 438 /v cm s

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

6

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L

Figure 3.3: Graph for Deflection of beam vs. Travelling time

for undamaged beam for 573 /v cm s

0 0.03 0.06 0.09 0.12 0.15 0.18-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L



39


1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

6

7

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3573 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18-1

0

1

2

3

4

5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L



40

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

6

7

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.5,0.65,0.85v cm s

0 0.03 0.06 0.09 0.12 0.15 0.18-1

0

1

2

3

4

5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L

Figure 3.7 Graph for Deflection of beam vs. Travelling time for

1,2,3 1,2,3573 / , 0.6,0.25,0.45. 0.5,0.65,0.85v cm s



41

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

6

7

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3438 / , 0.3,0.55,0.4. 0.25,0.45,0.65v cm s

0 0.04 0.08 0.12 0.16 0.2-1

0

1

2

3

4

5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.25,0.45,0.65v cm s



42

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

6

7

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3438 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s

0 0.03 0.06 0.09 0.12 0.15 0.18-1

0

1

2

3

4

5

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L

M=2kg, x=vt

M=2 kg, x=L


1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s



43

00.05

0.1

0.150.2

1

2

3

40

2

4

6

8

Time 't' in secMoving mass in 'kg'

Bea

m d

efle

ctio

n i

n 'c

m'(

x =

vt)

Figure 3.12: 3-D Graph for Deflection of beam vs. mass vs. Travelling time

for 1,2,3 1,2,3573 / , 0.3,0.55,0.4. 0.5,0.65,0.85v cm s

00.05

0.10.15

0.20.25

0

2

4

6-2

0

2

4

6

8

10

Time 't' in secMass in 'kg'

Def

lect

ion

of

bea

m i

n 'c

m'(

x =

L)

Figure 3.13: 3-D Graph for Deflection of beam vs. mass vs. Travelling time

for 1,2,3 1,2,3438 / , 0.6,0.25,0.45. 0.25,0.45,0.65v cm s



44

Figure 3.14: 3-D Graph for Deflection of beam vs. speed vs. Travelling time

for 1,2,3 1,2,32 , 0.6,0.25,0.45. 0.25,0.45,0.65M kg

00.05

0.10.15

0.20.25

0.30.35

300

400

500

600

7000

2

4

6

8

10

Time 't' in secSpeed in 'cm/s'

Beam

defl

ecti

on

in

'cm

' (x

=v

t)

Figure 3.15: 3-D Graph for Deflection of beam vs. Travelling time vs. Position

of mass for 1,2,3 1,2,32 , 573 / , 0.6,0.25,0.45. 0.25,0.45,0.65M kg v cm s



45

3.6 Theoretical-Numerical Solution of Cracked Simply

Supported Structure under a Moving Mass

A mass ‘ M ’is moving with a speed ‘ v ’ on a multi-cracked simply supported beam from

the supported left end to the supported right end as in Figure 3.16. The cracks are located

from the left support end of the structure. The responses of the cracked simply supported

under a moving mass are calculated using equation (3.18).The various end states of the

simply supported structure are as follows:

At 0, (0, ) 0, (0, ) 0x y t t and at , ( , ) 0, ( , ) 0x L y L t L t . (3.19)

A mild steel multi-cracked simply supported structure subjected to a traversing mass has

been considered for the numerical analysis of the following dimensions.

L= 140cm, B=4.9cm, H=0.5cm. M =1 kg and 2 kg. 1,2,3 0.2, 0.3, 0.4 and 0.35, 0.45,

0.55. v = 438 cm/s and 573 cm/s.

1,2,3 0.2857, 0.5, 0.7143 and 0.1786, 0.3571, 0.5714=Relative crack positions from the

left support end. The deflections of the cracked simply supported beam are calculated at

different positions of the moving mass along with the mid-span and analyzed in the

following Figures.

h



46

Figure 3.17: Graph for Deflection of beam vs. Travelling

time for undamaged beam for 438 /v cm s

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


undamaged beam for 573 /v cm s



47

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


for 1,2,3 1,2,3438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


for 1,2,3 1,2,3573 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s



48

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2

Figure 3.21 Graph for Deflection of beam vs. Travelling time

for 1,2,3 1,2,3438 / , 0.35,0.45,0.55. 0.2857,0.5,0.7143.v cm s

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3573 / , 0.35,0.45,0.55. 0.2857,0.5,0.7143.v cm s



49

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3438 / , 0.2,0.3,0.4. 0.1786,0.3571,0.5714.v cm s

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3573 / , 0.2,0.3,0.4. 0.1786,0.3571,0.5714.v cm s



50

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3438 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s

0 0.05 0.1 0.15 0.2 0.25-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3573 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s



51

Figure 3.28: 3-D Graph for Deflection of beam vs. speed. Travelling time

1,2,3 1,2,3 2 , 0.35,0.45,0.55. 0.1786,0.3571,0.5714M kg

00.1

0.20.3

0.4

300400

500600

700800-0.5

0

0.5

1

1.5


Defl

ecti

on o

f beam

in 'cm

' (x

=vt)

00.05

0.10.15

0.20.25

1

2

3

4

5-1

0

1

2

3

4


Defl

ecti

on

of

beam

in

'cm

'(x

= v

t)

Figure 3.27: 3-D Graph for Deflection of beam vs. mass vs. travelling time

for 1,2,3 1,2,3 573 / , 0.35,0.45,0.55. 0.1786,0.3571,0.5714v cm s



52

00.1

0.20.3

0.4

1

2

3

4

5-2

-1

0

1

2

3


Defl

ecti

on

of

beam

in

'cm

'(x

= L

/2)

Figure 3.29: 3-D Graph for Deflection of beam vs. mass. Travelling time

1,2,3 1,2,3 438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143v cm s

Figure 3.30: 3-D Graph for Deflection of beam vs. Travelling time vs. Position of

mass for 1,2,3 1,2,3 1 , 438 / , 0.2,0.3,0.4. 0.2857,0.5,0.7143M kg v cm s



53

3.7 Theoretical-Numerical Solution of Cracked Fixed-Fixed

Beam under a Moving Mass

A fixed-fixed structure with multiple cracks under a traversing mass ‘ M ’with speed ‘ v ’

is analysed as in Figure 3.31. The cracks are positioned from the fixed left part of the

structure. The responses of the multi-cracked fixed-fixed structure subjected to a transit

mass are determined by employing equation (3.18) with proper end conditions. For the

numerical analysis, a multi-cracked fixed-fixed beam of mild steel has been considered

with the prescribed dimensions.

L=140cm, B=4.9cm, H=0.5cm. M =1kg and 2kg. v = 512 cm/s and 617 cm/s. 1,2,3 0.2,

0.35, 0.45 and 0.3, 0.5, 0.55. 1,2,3 0.1429, 0.3214, 0.5357 and 0.25, 0.4286,

0.7143=Relative crack positions from the left fixed end.

1L

2L

x

h

3L

L

B

H

1,2,3d

1d

2d

3d

M v

Figure.3.31: Multi-cracked fixed-fixed beam subjected to moving mass



54

The deflections of the cracked fixed-fixed beam are determined at different positions of

the transit mass along with the mid-section of the beam and analyzed in the following

Figures.

0 0.05 0.1 0.15 0.2 0.25-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


for undamaged beam 617 /v cm s


for undamaged beam for 512 /v cm s

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2



55

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time 't' insec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357v cm s

0 0.05 0.1 0.15 0.2 0.25-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3617 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357v cm s



56

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3512 / , 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s

0 0.05 0.1 0.15 0.2 0.25-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3617 / , 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s



57

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3512 / , 0.3,0.5,0.55. 0.1429,0.3214,0.5357v cm s

0 0.05 0.1 0.15 0.2 0.25-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


1,2,3 1,2,3617 / , 0.3,0.5,0.55. 0.1429,0.3214,0.5357v cm s



58

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


for 1,2,3 1,2,3512 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143v cm s

0 0.05 0.1 0.15 0.2 0.25-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

M=1kg, x=vt

M=1kg, x=L/2

M=2kg, x=vt

M=2kg, x=L/2


for 1,2,3 1,2,3617 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143v cm s



59

Figure 3.43: 3-D Graph for Deflection of beam vs. speed vs. Travelling

time for 1,2,3 1,2,3 2 , 0.3,0.5,0.55. 0.25,0.4286,0.7143.M kg

00.05

0.10.15

0.20.25

0.30.35

500

600700

800

9001000

-1

-0.5

0

0.5

1


Defl

ecti

on o

f beam

in 'cm

' (x

=L

/2)

00.05

0.10.15

0.20.25

1

2

3

4

5-2

-1

0

1

2


Def

lect

ion

of

bea

m i

n 'c

m' (

x =

L/2

)

Figure 3.42: 3-D Graph for Deflection of beam vs. mass vs. Travelling

time for 1,2,3 1,2,3 617 / , 0.3,0.5,0.55. 0.25,0.4286,0.7143.v cm s



60

00.1

0.20.3

0.4

1

2

3

4

5-0.5

0

0.5

1

1.5


Def

lect

ion

of

bea

m i

n 'c

m'(

x =

vt)

Figure 3.44: 3-D Graph for Deflection of beam vs. mass vs. Travelling

Time for 1,2,3 1,2,3 512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357.v cm s

Figure 3.45: 3-D Graph for Deflection of beam vs. Travelling time vs. Position of

mass for 2 ,M kg 1,2,3 1,2,3 512 / , 0.2,0.35,0.45. 0.1429,0.3214,0.5357.v cm s



61

3.8 Identification of cracks form the measured dynamic

response of structures

The graphs between the relative distance and relative deflections of structures under the

transit mass are shown in Figures 3.46 (Cracked cantilever beam), 3.47 (Cracked simply

supported beam) and 3.48 (Cracked fixed-fixed beam). The magnified views of cracks for

the case of cantilever beam are presented in Figures 3.49(a), (b), (c) at different relative

crack locations. It has been observed that abrupt changes in relative deflections occurred in

the dynamic response of structures. The sudden changes in relative deflections of structures

will predict the existence, locations and severities of cracks.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

Relative distance from fixed end

Rel

ativ

e d

efle

ctio

n

Location of first crack

Location of second crack

Location of third crack

Figure 3.46: Detection of cracks for cantilever beam for 1,2,3 0.5,0.65,0.85



62

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Relative distance from left support end

Rel

ativ

e d

efle

ctio

n




Figure 3.47: Detection of cracks for simply supported beam for

1,2,3 0.2857,0.5,0.7143.

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Relative distance from left fixed end

Rel

ativ

e d

efle

ctio

n




Figure 3.48: Detection of cracks for fixed-fixed beam for

1,2,3 0.25,0.4286,0.7143.



63

0.04

0.06

0.08

0.1

0.12

0.485 0.49 0.495 0.5 0.505 0.51 0.515

Rel

ati

ve

def

lect

ion


Undamaged

Damaged

Figure 3.49(a): Magnified view of crack for 0.5

Figure 3.49(b): Magnified view of crack for 0.65

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.635 0.64 0.645 0.65 0.655 0.66 0.665

Rel

ati

ve

def

lect

ion


Undamaged

Damaged



64

3.9 Comparison of Results of Theoretical-Numerical and

Experimental analysis for the Response of Structures

For the validation and accuracy of the applied numerical method (Runge-Kutta method),

experimental verifications are carried out in the laboratory. Detailed analyses of the

experimental verifications have been elaborated in Chapter-8. The experiment has been

conducted with different types of structures (cantilever, simply supported and fixed-fixed

beam) under moving mass with some number of observations. For the laboratory tests, the

same configurations as those of numerical models are considered with the same beam

specimen and dimensions. The deflections of the beam are measured at different positions

of the moving mass and the specified location of the beams during the movement of the

traversed mass on the beam. The results obtained from the computational analysis, and

laboratory tests are shown in the below Tables for comparison studies. The variations of

results, time (sec) ~deflection (cm) for different structures under transit mass, obtained

from both the computational analyses and laboratory investigation are given in Tables 3.1

(Cracked cantilever beam), 3.2 (Cracked simply supported beam) and 3.3 (Cracked fixed-

fixed beam) for the comparison of results. It has been observed that the variation of results

of the numerical analysis and experimental verification are within the estimated error of

5% approximately for all the structures. So the applied numerical method (Runge-Kutta

method) yields well with the experimental verifications.

Figure 3.49(c): Magnified view of crack for 0.85

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.835 0.84 0.845 0.85 0.855 0.86 0.865

Rel

ati

ve

def

lect

ion


Undamaged

Damaged



65

The percentage of deviation between the experimental and numerical values is given by

the relation= (Experimental Numerical)

100Experimental

Average percentage of error= Sum of the percentage errors

Total number of observations

Total percentage of error = Sum of the average percentage errors

Total number of average percentage of errors

Table 3.1: Comparison of results for beam deflection (cm) between experiment and numerical for

cracked cantilever beam for 1,2,3 1,2,3438 / . 0.6,0.25,0.45. 0.25,0.45,0.65.v cm s

Time

(sec)

Numerical

(x=vt)

Numerical

(x=L)

Experiment

(x=vt)

Experiment

(x=L)

M=1 kg M=2 kg M=1 kg M=2 kg M=1 kg M=2 kg M=1 kg M=2 kg

0.0342 0.0061 0.0118 -0.0061 -0.0115 0.0063 0.012 -0.0062 -0.0117

0.04 0.0101 0.0196 -0.0021 -0.0035 0.0103 0.02 -0.0023 -0.0036

0.0571 0.0287 0.0553 0.0459 0.0891 0.0295 0.0569 0.0471 0.092

0.0799 0.1055 0.1905 0.2215 0.4328 0.1089 0.1976 0.2288 0.4512

0.1027 0.2379 0.4299 0.6629 1.2206 0.2484 0.4502 0.6921 1.2787

0.1256 0.5359 0.917 1.2909 2.303 0.567 0.97 1.3641 2.4254

0.1484 1.0222 1.6768 1.9606 3.1873 1.0783 1.7849 2.0897 3.3663

0.1769 2.0423 3.2041 2.9394 4.6597 2.1698 3.4386 3.1487 4.9959

0.1988 3.0302 4.6314 3.6836 5.6087 3.2384 4.9816 3.9783 6.0595

0.2169 3.9061 5.8909 4.2145 6.3633 4.2342 6.3844 4.5929 6.9286

Average percentage of errors 4.56 4.84 4.85 4.98

Total percentage of error 4.8



66


cracked simply supported beam for 1,2,3 1,2,3573 / . 0.2,0.3,0.4. 0.2857,0.5,0.7143.v cm s

Time

(sec)

Numerical

(x=vt)

Numerical

(x=L/2)

Experiment

(x=vt)

Experiment

(x=L/2)


0.0428 0.0414 0.0785 0.0496 0.094 0.042

0.08 0.0505 0.0962

0.0672 0.1297 0.2383 0.1579 0.2942 0.1346 0.2454 0.1627 0.3049

0.0855 0.2458 0.4533 0.2604 0.4776 0.2552 0.4749 0.27 0.5013

0.1038 0.3538 0.6606 0.3614 0.6739 0.3731 0.6977 0.3859 0.7157

0.1222 0.4357 0.8408 0.4357 0.8408 0.4663 0.9103 0.4717 0.901

0.1466

0.4889 1.023 0.4703 0.9842 0.5302 1.117 0.5133 1.0679

0.171 0.3807 0.9143 0.4146 0.9894 0.4099 0.9815 0.4451 1.0788

0.1894 0.2632 0.7227 0.3059 0.8567 0.2798 0.763 0.3247 0.9126

0.2077 0.104 0.3622 0.1605 0.5901 0.1090 0.3751 0.1676 0.6186

0.2199 0.0242 0.1233 0.0251 0.34 0.0537 0.1265 0.0559 0.3507




cracked fixed-fixed beam for 1,2,3 1,2,3617 / . 0.2,0.35,0.45. 0.25,0.4286,0.7143.v cm s

Time

(sec)

Numerical

(x=vt)

Numerical

(x=L/2)

Experiment

(x=vt)

Experiment

(x=L/2)


0.034 0.0113 0.0215 0.0165 0.0319 0.0115 0.0219 0.017 0.0326

0.0567 0.0541 0.1012 0.0744 0.1388 0.0554 0.1043 0.0766 0.1432

0.0794 0.12 0.2331 0.1388 0.2687 0.1248 0.2525 0.1444 0.2809

0.1021 0.1759 0.369 0.1772 0.3713 0.184 0.3899 0.1861 0.3927

0.1248 0.1892 0.4245 0.1907 0.4282 0.1997 0.4557 0.2025 0.4593

0.1418 0.169 0.395 0.1778 0.4154 0.1805 0.4295 0.1903 0.4509

0.1588 0.1172 0.2829 0.1334 0.3241 0.1269 0.3048 0.1441 0.3468

0.1759 0.0577 0.1236 0.0652 0.1515 0.0612 0.1314 0.0689 0.1599

0.1929 0.0136 -0.0075 0.0117 -0.0432 0.0144 -0.0078 0.0123 -0.0453

0.2042 0.0039 -0.006 0.0034 -0.1136 0.0041 -0.0062 0.0035 -0.1175





67

3.9 Discussions and Summary

The solutions to the problem for the response of structures under transit mass have been

found out by applying Runge-Kutta fourth order rule. The solution can be implemented for

any types of the structure under transit mass with various boundary conditions. The

governing equation of motion of the dynamic mass-structural systems has been addressed

as a sequence of fixed partial differential equations. It is always assumed that there is no

separation between the moving mass and the structure. The deflections of the beam at each

location of the transit mass on the beam throughout the movement of the transit mass along

the structure and the tip end deflection (cantilever beam), and mid-span deflection (simply

supported and fixed-fixed beam) of the structure have been also calculated during the

movement of the mass across the beam at different mass, speed, crack depth and crack

locations. Numerical studies have been illustrated for various types of beam structures

subjected to transit mass for determining the deflections of the structures. From the analysis

of Figures 3.2-3.11, 3.17-3.26 and 3.31-3.41, it has been observed that the deflection

induced due to the moving mass of a damaged beam is greater than that of the undamaged

beam. Again by increasing the crack depth and weight of the traversing mass, the

deflections of the structure also increases.

In the case of a cantilever beam structure (Figures 3.2-3.11); the deflections at each location

of the moving mass ( )x vt on the beam and the tip end ( )x L of the beam are evaluated

during the movement of the transit mass on the structure. It has been observed that the tip

end deflection of the structure decreases with the enhancement of the velocity of the transit

mass. This is because, at greater velocity of the transit mass, the lower modes of the

structures are not effective and hence less deflection is observed in the structure. However

the transverse dynamic deflections of the beam mainly depend on the vibration of the lower

modes of the structure. It has also been noticed that the deflections of the beam at the tip

end reduces sharply with time and again amplify at the higher speed of the moving mass,

it’s because vibration at superior modes is more prevailing than those of inferior modes.

The vibrant deflection at the tip end of the cantilever structure depends upon the modal

excitation of the structure. So the deflection at the tip end shows sudden variation in

magnitude with direction. If the location of cracks moves towards the fixed end of the

cantilever structure, then the corresponding beam deflection also amplifies.



68

In the case of a simply supported beam structure, the mass is assumed to move from the left

to right. The dynamic deflections of the structure at each location of the moving mass

( )x vt on the structure and the middle ( )2

Lx of the structure are evaluated during the

passage of the transit mass along the structure. From the analysis of Figures 3.17-3.26, it

has been noticed that the dynamic deflection at the position of the moving mass ( )x vt on

the structure progressively amplifies till the mass reaches at the mid-span of the structure,

then the deflections of the structure start reducing towards the right end. But the beam mid-

span deflection ( )2

Lx initially increases till the mass reaches the mid-span, starts

reducing towards the right end. With the amplification of the traversing speed, the

deflection of the structure ( )x vt decreases up to the mid-span of the beam, then after

crossing the mid-span, it again amplifies towards the right end. However with the

amplification of speed, the deflection at the beam mid-span ( )2

Lx decreases until the

mass travels to the mid of the structure, then it starts increasing on the way to the right end.

If the crack locations approach towards the supported right end, then the displacement of

the beam decreases and increases somewhat towards the right support end.

In the case of the fixed-fixed beam, the mass is assumed to move from the fixed left end to

right end. The responses of the structure under transit mass have been analyzed in Figures

3.31-3.41. It has been noticed that the deflection of the fixed-fixed structure at the position

of the transit mass ( )x vt on the structure gradually amplifies till the traversing object

attains the middle of the structure, then the deflections start reducing towards the right end

of the structure. While the beam mid-span deflection ( )2

Lx primarily increases till the

mass traverses to middle of the beam, then it decreases gradually after the passage of the

mid-span and commence to drop quickly towards the right end of the structure. With the

enhancement of the weight of the transit mass, the deflections of the

structure ( and )2

Lx vt x gradually increase, but while approaching towards the right

end of the structure, it decreases up to some extents of the structure and then it increase

towards the end. With the amplification of the speed of the transit mass, the beam

deflection ( and )2

Lx vt x primarily decreases, and then an increase approaching till the

mass reaches at the mid-span of the beam. The beam deflections decreases while the



69

traversing mass approaches towards the end of the structure. If the location of cracks

approach towards the fixed right end of the structure, then the deflection produced is less.

3-D graphs are plotted and explained in Figures 3.12-3.15 (cracked cantilever beam),

Figures 3.27-3.30(cracked simply supported beam), and Figure 3.42-3.45 (cracked fixed-

fixed beam) for various masses, position of the transit mass and speed. The variation of

deflections are obtained at different positions of the transit mass on the structure ( )x vt ,

end point of the cantilever beam ( )x L , mid-span of both simply supported and fixed-

fixed beam ( )2

Lx for variation in different weights, velocities of the transit mass. The

responses of the damaged structures are also studied from the behaviour of the 3-D plots.

Similar observations are also obtained from the plot of 3-D graphs.

The laboratory tests are conducted for all the structures at different configurations of mass,

speed, relative crack depth and crack locations. The variation of results obtained from both

numerical studies and laboratory tests are illustrated in Tables 3.1,2 and 3. The results

obtained from the numerical studies agree well with experimental results and the deviation

is to be within 5%. So the applied numerical method yields well. The magnified views of

the cracks for the case of cantilever beam are shown in Figures 3.49 (a), (b) and (c). From

the measured dynamic response of the structures, the feasible existence, locations and

intensities of cracks can be predicted.

The beam displacements at any position of the transit mass on the structure have mainly

depended on the weight of the transit mass. It has been concluded that parameters like the

weight and speed of the transit mass, depth and location of cracks affect the dynamic

behaviour and response of the structure.

70

Chapter 4

FINITE ELEMENT ANALYSIS

CRACKED STRUCTURES SUBJECTED

TO MOVING MASS

4.1 Introduction

The finite element methods (FEM) with nonlinearities are frequently employed to find the

solutions to problems related to engineering structures. The competent supervision of

bridges and highway structures, where the information based on the condition of

structures, real effects of dynamic loading, the impact of speed and mass on the dynamic

characteristics of structures are various issues for preparing administration assessment and

establishing the load limit of the structures. Finite Element Analysis (FEA) can be applied

to explain the real response of structures due to moving load. The FEA of moving mass-

structure interaction dynamic was carried out using the commercial ANSYS

WORKBENCH 2015 in this Chapter. The results from the FEA are verified with those of

experiments and numerical formulations to ensure the accuracy and exactness of the

proposed analysis.

4.2 Method for FEA of moving mass-structure using

ANSYS

In ANSYS, the FEA of transit mass-structure interaction dynamic is carried out by using

transient dynamic analysis method. The transient dynamic analysis is a computational

method that is applied to evaluate the response of the structure under moving load varying

in time. This method is useful for calculating the time-varying strains, displacements and

stresses of structures. The general non-linear principal equation of motion for the transient

dynamic analysis is expressed as-

[ ] [ ] [ ] ( )t t tM x C x K x f t (4.1)

Chapter 4 Finite Element Analysis Cracked Structures Subjected to Moving Mass

71

Where [ ]tM x -Inertial force, [ ]tC x -Damping force, [ ]tK x -Stiffness force, ( )f t -Applied

force. The schematic view of the vibrating system under applied force has been shown in

Figure 4.1.

To find out the solution of equation (4.1), time integration has to be carried out. The

implicit time integration method has been chosen for the present investigations. In

ANSYS, the implicit time integration technique is Newmark’s integration method. In

ANSYS Mechanical, two types of solution methods (The full method and the mode

superposition method) can be applied to find out the time-varying responses of the

structure.

The full method: This method is a simple one to set up. The entire matrices [M, C and

K] are applied to evaluate the stresses, displacements and strains in a single pass. All kinds

of nonlinearities, efficient employment of solid-model loads and regular time stepping are

allowed in this analysis. The principal drawback of this method is the requirement of more

computational time for the size of the prescribed model.

The mode superposition method: The fundamental concept of this method is to explain

the responses of a structure to the linear combination of its all undamped mode shapes.

This approach is quicker and requires less computational time than that of the ‘The full

method’. Damping is allowed as in terms of frequency. The step of time is fixed. The

disadvantage of this method is that the nonlinearities are not allowed.

The full method, transient dynamic analysis, is applied in the present analysis due to the

inclusion of nonlinearities. In this analysis, Newmark’s time integration method under

zero damping conditions has been applied to evaluate the response of structures in

K C

U

M

Figure 4.1: Free body diagram of vibrating system


72

ANSYS WORKBENCH 2015 domain. Newmark’s integration parameters under the

schemes of unconditionally stable and constant average acceleration method are

considered in this analysis.

4.3 Steps involving ‘The full method’ transient dynamic

analysis in ANSYS

Before analysing the different steps of transient dynamic analysis, modal analysis is to be

completed to extract the mode shapes and natural frequencies of the structures.

Step 1- Select Transient structural

Step 2- Select Material properties (material types, ρ, E, υ, k, G).

Step 3- Build the geometry

Step 4- Select Model-Contact (No separation) - Sliding is allowed

Step 5- Mesh – (Element size)

Step 6- Select Transient- Initial conditions

Step 7- Analysis setting- Give the ending time of step

-Automatic time stepping is allowed (on)

-Initial step timing = 120

-Set minimum step timing

-Set maximum step timing

-Time integration (on)

-Standard earth gravity (g)

-End conditions

-Speed

Step 8- Select solution- Directional deformation

Here ‘ ’ is the highest natural frequency of the structure.

4.4 Response analysis of cracked structures under moving

mass using ANSYS

The FEA for the dynamic response of structures subjected to transit mass has been

performed using the commercial ANSYS WORKBENCH 2015. In the primary stage,

modal analyses up to three modes of vibration are performed to extract the mode shapes

and the natural frequencies of the structures.


73

In ANSYS, the responses of the structures at different locations of the transit mass and the

particular location of the structure are calculated. The dimensions of the structures are

same as those of experimental model with the same damage configurations, traversing

mass and speed. The interaction of transit mass-structure dynamic for the case of the

Figure 4.2 Transit mass-structure interaction of cracked cantilever beam for

1,2,3 1,2,30.3,0.55,0.4. 0.5,0.65,0.85. M=2 kg

Figure 4.3: Magnified view of crack for α=0.55


74

cantilever structure is shown in Figure 4.2. The crack modelling is also carried out in

ANSYS. The magnified view of a crack is shown in Figure 4.3.

Table 4.1: Frequencies ratios of damaged cantilever beam

Mode

No

1,2,3 0.6,0.25,0.45.

1,2,3 25,45,65L cm

1,2,3 0.3,0.55,0.4.

1,2,3 50,65,85 .L cm

1,2,3 0.6,0.25,0.45.

1,2,3 50,65,85 .L cm

1,2,3 0.3,0.55,0.4.

1,2,3 25,45,65L cm

1 0.9908 0.9971 0.9674 0.9843

2 0.9587 0.9728 0.9833 0.9662

3 0.9904 0.9718 0.9638 0.9824

Length of the beam

Def

orm

atio

n o

f th

e bea

m

Figure 4.4 (a): Second mode shape of cantilever structure


75

Figure 4.4 (b): Third mode shape of cantilever structure

Length of the beam

Def

orm

atio

n o

f th

e b

eam

Figure 4.5: Schematic view of transient structural model for cracked cantilever beam


76

Table 4.2: Comparison of results for beam deflection (cm) between experiment and FEA for

cracked cantilever beam for 1,2,3 1,2,3573 / . 0.3,0.55,0.4. 0.5,0.65,0.85.v cm s

Time

(sec)

Experiment

(x=vt)

Experiment

(x=L)

FEA

(x=vt)

FEA

(x=L)


0.0262 0.0056 0.0105 -0.0075 -0.0146 0.0055 0.0103 -0.0074 -0.0144

0.0436 0.0289 0.054 0.0073 0.015 0.0281 0.0529 0.0072 0.0147

0.0611 0.0687 0.1291 0.1144 0.2163 0.0665 0.1256 0.1116 0.2114

0.0785 0.1444 0.2618 0.3646 0.6966 0.1394 0.2541 0.3542 0.6788

0.096 0.3309 0.5724 0.7786 1.4428 0.3189 0.5536 0.7546 1.4009

0.1134 0.6791 1.124 1.290 2.1807 0.6535 1.0857 1.2496 2.1101

0.1309 1.176 1.8549 1.866 2.9291 1.1313 1.7894 1.8024 2.8292

0.144 1.6694 2.5523 2.3412 3.6142 1.6044 2.4581 2.2574 3.4877

0.1571 2.4058 3.5935 2.7968 4.1889 2.3115 3.4548 2.6933 4.0343

0.1658 2.8857 4.2526 3.1063 4.5708 2.7712 4.0834 2.9858 4.3903


Total average percentage of error 3.13

The percentage of error between the experimental and FEA values are given by the

following relation, Percentage of error= (Expt.values-FEA values)100

Expt.values .

Average percentage of error= Sum of the percentage errors

Total number of observations

Total percentage of error = Sum of the average percentage errors

Total number of average percentage of errors


77

Table 4.3: Comparison of results for beam deflection (cm) among experiment, FEA and numerical

for cracked cantilever beam for M=1kg,v=438cm/s, 1,2,3 1,2,30.3,0.55,0.4. 0.5,0.65,0.85

Time

(sec)

Experiment FEA

Numerical

(x=vt) (x=L) (x=vt) (x=L) (x=vt) (x=L)

0.0457 0.0155 0.0073 0.0152 0.0072 0.0151 0.0071

0.0685 0.0515 0.1117 0.0502 0.1096 0.0496 0.1082

0.0913 0.1295 0.3818 0.1257 0.3727 0.124 0.3675

0.1142 0.2893 0.8606 0.2802 0.8362 0.2759 0.8213

0.137 0.6626 1.4647 0.6411 1.4171 0.6297 1.3894

0.1541 1.091 2.0048 1.0549 1.9378 1.0348 1.8962

0.1712 1.5984 2.5734 1.5441 2.4837 1.5111 2.4244

0.1884 2.2184 3.137 2.1417 3.0175 2.0929 2.9412

0.2055 3.1476 3.7036 3.0372 3.5544 2.9634 3.4606

0.2226 4.0492 4.2408 3.9023 4.0636 3.8028 3.9474


Total average percentage of error 4.03

0 0.05 0.1 0.15 0.2 0.25-1

0

1

2

3

4

5

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

Numerical, x=vt

FEA, x=vt

Expt., x=vt

Numerical, x=L

FEA, x=L

Expt., x=L

Figure 4.6: For cracked cantilever beam for 1 , 438 / ,M kg v cm s

1,2,3 1,2,30.3,0.55,0.4. 0.25,0.65,0.85


78

For the cantilever structure, before analysing the transient dynamic analysis, the mode

shapes and the natural frequencies of the structure are determined with different damage

configuration of the structure by considering the first three modes of vibration. The

proportions of the natural frequencies for cantilever beam are presented in Table 4.1 with

various damage configuration of the structure. The different mode shapes of the cantilever

are shown in Figures 4.4(a) and (b). The responses of the damaged cantilever beam are

found out using the full method transient dynamic analysis. The schematic view of the

transient structural model in ANSYS domain is explained in Figure 4.5 for the cracked

cantilever beam. In ANSYS, Figure 4.5, the symbol, M(y), presents the deflections of the

structures at the locations of the transit mass, and L(y) represents deflections of the

structure at the free end. The results from the experimental verification are compared with

those of FEA and numerical for the validation of the FEA method. The comparison of

results, time (sec) ~ deflections (cm), between experiments and FEA are presented in

Table 4.2, and those of among experiments, FEA and numerical are presented in Table

4.3. It has been observed that the experimental results agree well with those of result from

FEA with an average error about 3%. So the applied integration method in ANSYS,

Newmark integration method, converges well with the experiment. So the applied method

in FEA is a valid one.

Length of the beam

Def

orm

atio

n o

f th

e bea

m

Figure 4.7(a) Second mode shape of simply supported beam


79

Table 4.4: Frequencies ratios of damaged simply supported beam Mode

No 1,2,3 0.2,0.3,0.4.

1,2,3 40,70,100L cm

1,2,3 0.35,0.55,0.45.

1,2,3 40,70,100L cm

1,2,3 0.2,0.3,0.4.

1,2,3 25,50,80L cm

1,2,3 0.35,0.55,0.45.

1,2,3 25,50,80L cm

1 0.9816 0.9647 0.9844 0.9622

2 0.9907 0.9795 0.9947 0.9875

3 0.9921 0.9812 0.9952 0.9866


cracked simply supported beam for 1,2,3 1,2,3438 / . 0.35,0.45,0.55. 0.1786,0.3571,0.5714.v cm s

Time

(sec)

Experiment

(x=vt)

Experiment

(x=L/2)

FEA

(x=vt)

FEA

(x=L/2)


0.0479 0.037 0.0708 0.0628 0.1204 0.0365 0.0698 0.0618 0.1188

0.0799 0.1664 0.3097 0.219 0.4067 0.1636 0.3038 0.2148 0.4001

0.1039 0.2945 0.5578 0.3641 0.6775 0.2889 0.545 0.3562 0.6641

0.1279 0.4624 0.8934 0.4752 0.9142 0.4518 0.8703 0.4635 0.8943

0.1518 0.54 1.1024 0.5428 1.105 0.5257 1.0699 0.5278 1.0767

0.1758 0.5387 1.1856 0.5489 1.2108 0.5226 1.146 0.5295 1.1747

0.2078 0.4672 1.1647 0.4577 1.1462 0.4524 1.1227 0.4399 1.1068

0.2317 0.3309 0.882 0.3541 0.9723 0.3191 0.8471 0.3411 0.9344

0.2557 0.1937 0.5035 0.2389 0.6515 0.1866 0.4885 0.2308 0.629

0.2797 0.0785 0.1280 0.1368 0.2528 0.0755 0.1246 0.1327 0.2448



Length of the beam

Def

orm

atio

n o

f th

e b

eam

Figure 4.7(b) Third mode shape of simply supported beam


80


for cracked simply supported beam for M=1kg, v=573cm/s,

1,2,3 1,2,30.35,0.45,0.55. 0.1876,0.3571,0.5714.

Time

(sec)

Experiment FEA

Numerical

(x=vt) (x=L/2) (x=vt) (x=L/2) (x=vt) (x=L/2)

0.0428 0.0425 0.0585 0.0418 0.0576 0.0414 0.0569

0.0733 0.1913 0.2327 0.1876 0.2281 0.1852 0.2248

0.0977 0.3998 0.4124 0.3897 0.4035 0.3836 0.3969

0.1161 0.5378 0.5374 0.522 0.5225 0.5124 0.5129

0.1344 0.6226 0.6263 0.6028 0.6067 0.589 0.5935

0.1527 0.6889 0.659 0.6643 0.6357 0.6479 0.6199

0.171 0.5948 0.6259 0.5719 0.6019 0.5567 0.5852

0.1894 0.4125 0.5053 0.3952 0.4898 0.3841 0.4755

0.2077 0.1759 0.3069 0.1707 0.298 0.1672 0.2919

0.226 0.0157 0.0483 0.0154 0.0471 0.0152 0.0463



Table 4.7: Frequencies ratios of damaged fixed-fixed beam Mode

No 1,2,3 0.3,0.5,0.55.

1,2,3 20,45,75L cm

1,2,3 0.2,0.35,0.45.

1,2,3 35,60,100L cm

1,2,3 0.3,0.5,0.55.

1,2,3 35,60,100L cm

1,2,3 0.2,0.35,0.45.

1,2,3 20,45,75L cm

1 0.9834 0.9949 0.9893 0.9898

2 0.9852 0.9878 0.9794 0.9942

3 0.9815 0.9902 0.9817 0.9881


81

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time 't' in sec

Defl

ecti

on

of

beam

in

'cm

'

Numerical, x=vt

FEA, x=vt

Expt., x=vt

Numerical, x=L/2

FEA, x=L/2

Expt., x=L/2

Figure 4.8: For cracked simply supported beam for 2 , 438 /M kg v cm s ,

1,2,3 1,2,30.35,0.45,0.55. 0.2857,0.5,0.7143 .

Length of the beam

Def

orm

atio

n o

f th

e bea

m

Figure 4.9(a) Second mode shape of fixed-fixed beam


82


cracked fixed-fixed beam for 1,2,3 1,2,3512 / . 0.3,0.5,0.55. 0.25,0.4286,0.7143.v cm s

Time

(sec)

Experiment

(x=vt)

Experiment

(x=L/2)

FEA

(x=vt)

FEA

(x=L/2)


0.0547 0.0325 0.0622 0.0544 0.1043 0.032 0.061 0.0536 0.1025

0.082 0.0955 0.1877 0.1177 0.2327 0.0936 0.1836 0.1154 0.2271

0.1094 0.1517 0.3136 0.1757 0.3636 0.1482 0.3046 0.1714 0.353

0.1299 0.2034 0.4308 0.203 0.4338 0.1975 0.4168 0.1969 0.419

0.1504 0.2271 0.4804 0.2264 0.4866 0.2198 0.4629 0.2192 0.4687

0.1709 0.2096 0.4553 0.2151 0.4815 0.2019 0.4373 0.2077 0.4627

0.1982 0.1185 0.2771 0.1027 0.3026 0.1149 0.2685 0.0988 0.2933

0.2188 0.0633 0.0672 0.0585 0.0739 0.0616 0.0655 0.0567 0.0721

0.2324 0.0258 0.0053 0.0334 -0.0568 0.0252 0.051 0.0325 -0.0554

0.2461 0.0097 0.0104 0.038 -0.0325 0.0094 0.0102 0.0372 -0.0319



Length of the beam

Def

orm

atio

n o

f th

e b

eam

Figure 4.9(b) Third mode shape of fixed-fixed beam


83


for cracked fixed-fixed beam for M=2kg,

v=617cm/s, 1,2,3 1,2,30.3,0.5,0.55. 0.25,0.4286,0.7143.

Time

(sec)

Experiment FEA

Numerical

(x=vt) (x=L/2) (x=vt) (x=L/2) (x=vt) (x=L/2)

0.0511 0.0772 0.1183 0.0762 0.1166 0.0751 0.1152

0.0681 0.1803 0.221 0.1775 0.217 0.1747 0.2137

0.0851 0.2864 0.3362 0.2808 0.3291 0.2754 0.3231

0.1078 0.4489 0.4489 0.4383 0.4379 0.4285 0.4289

0.1248 0.5043 0.5086 0.4908 0.4947 0.4783 0.4823

0.1361 0.5049 0.5235 0.489 0.5074 0.475 0.4924

0.1532 0.4227 0.4637 0.4060 0.4478 0.3936 0.4337

0.1702 0.2507 0.2943 0.2418 0.2831 0.2352 0.2739

0.1872 0.0269 0.0323 0.0261 0.0312 0.0255 0.0305

0.2156 -0.0026 -0.1106 -0.0025 -0.1072 -0.0024 -0.105



In the case of the damaged simply supported and fixed-fixed beam under transit mass, the

similar procedure like cantilever beam has been carried out to find the response of the

0 0.05 0.1 0.15 0.2 0.25-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time 't' in sec

Def

lect

ion

of

bea

m i

n 'c

m'

Numerical, x=vt

FEA, x=vt

Expt., x=vt

Numerical, x=L/2

FEA, x=L/2

Expt., x=L/2

Figure 4.10: For cracked fixed-fixed beam for 2 , 617 / M kg v cm s ,

1,2,3 1,2,30.2,0.35,0.45. 0.1429,0.3214,0.5357. .


84

structure in ANSYS domain. The frequency ratios of the structures at different damage

configuration are presented in Tables 4.4 (damaged simply supported beam) and

4.7(damaged fixed-fixed beam). The different mode shapes of the cracked structures are

shown in Figures 4.7(simply supported beam) and 4.9(fixed-fixed beam). The responses

of the structures are determined at each location of the transit mass and mid-span of the

structure at various damage scenarios of the structures. The results from the experiments

are compared with those of FEA and numerical analysis for the validation of the proposed

FEA scheme. It has been remarked that the error between the results of experimental

analysis and FEA are near about 2.7% for both the simply supported and fixed-fixed

structures.

4.5 Discussions and Summary

The FEA of the cracked structures subjected to transit mass has been carried out applying

the full method transient dynamic analysis in ANSYS WORKBENCH 2015 domain. The

computational method analysed in the full method transient dynamic analysis in ANSYS

domain is the Newmark-time integration method. Before analysing the transient dynamic

analysis, modal analyses are carried out for all the structures up to the first three modes of

vibration. The frequencies ratios of the multi-cracked structures are presented in Tables

4.1 (cracked cantilever beam), 4.4 (cracked simply supported beam), 4.7 (cracked fixed-

fixed beam) at various damaged configurations of the structures. The different mode

shapes of the damaged structures are shown in Figures 4.4(a, b) for cantilever structure,

Figures 4.7(a, b) for simply supported beam and Figures 4.9 (a, b) for the fixed-fixed

structure. In FEA, the responses of the structures have been found out at different

positions of the transit mass and the desired positions of the structure during the passage

of the transit mass across the structures. The experimental results are compared with those

of FEA in Tables 4.2(cracked cantilever beam), Table 4.6(cracked simply supported

beam) and Table 4.8 (cracked fixed-fixed beam). The comparison of results among

experiments, FEA and numerical analyses are also explained for damaged cantilever beam

(Table 4.3, Figure 4.6), simply supported beam (Table 4.6, Figure 4.8) and fixed-fixed

beam (Table 4.9, Figure 4.10). The errors between the results of numerical analyses and

FEA are about 1.9 % for cracked cantilever beam, 1.97% for cracked simply supported

beam and 2.36% for fixed-fixed beam. Regarding the response of the structures under

transit mass, similar observations are obtained in FEA like numerical and experimental


85

observations. It has been observed that results from the experiments are converged well

with the FEA with error about 3% for cantilever beam structure and 2.7% for the simply

supported and fixed-fixed beam structures. So the applied numerical method in ANSYS,

(Newmark’s time integration method), is an appropriate one to study the response of any

kind of structures under time-varying load.

86

Chapter 5

APPLICATION OF RECURRENT

NEURAL NETWORKS FOR DAMAGE

IDENTIFICATION IN STRUCTURES

UNDER MOVING MASS

5.1 Introduction

The features of all the real world structures are the realities that these are susceptible to

faults, natural calamities, breakdown, and more in general, unpredicted means of

performance. So they need a continuous system for consistent and complete monitoring of

structures based on efficient and appropriate fault diagnosis approach. This is happening

for engineering structures, whose complication is rising due to the mechanized

progression, unavoidable expansion of new industry along with the growing information

and technologies. The real design and safe operation of engineering structures focus on the

consistency, accessibility, fault tolerance and safety. So, it is the usual process for fault

diagnosis of structures in up to date control theory and practice. As a result, verities of

fault diagnosis methods arise for the better condition monitoring of structures.

Applications of artificial neural networks (ANN) are widely studied for the last two

decades and employed to fault diagnosis in structure along with the problems on

modelling of the dynamic system. ANNs present an exciting and precious option to

traditional approaches because the most complex problems are not adequately described

for the execution of deterministic algorithms. ANNs provide an exceptional mathematical

tool to analyze the non-linear problems. The features of ANNs are quiet useful for solving

problems in pattern recognition and their abilities for self-learning. In this Chapter,

recurrent neural networks (RNNs) are applied for damage identification in structures

subjected to transit mass. The Jordan’s RNNs, Elman’s RNNs, and the hybridization of

Chapter 5 Application of Recurrent Neural Networks for Damage Identification

in Structures under Moving Mass

87

the Jordan’s and Elman’s RNNs are employed for the fault detection of transit mass-

structure interaction problems.

5.2 Overview of neural networks

The neural networks can extort the features of the structure from the chronological

training data by the learning mechanism, knowing little or no previous information about

the method. This process gives the non-linear modelling of structures with greater

flexibility. The adaptive control structures are designed for unidentified, intricate and

non-linear dynamic procedures. The NNs can also work out strongly even if in the

presence of missing and incorrect data. The defensive imparting based on ANNs is also

not influenced by an alteration in the structure operating conditions. The NNs are also

capable for massive input error tolerance, significant computation rates, and adaptive

potential. Based on the training processes, the ANNs are classified into two broad

categories i.e. feed forward neural network (FFNNs) and recurrent neural networks

(RNNs).

5.2.1 Feed forward neural networks

The simplified structure of a feed forward ANN with multi-inputs and multi-outputs is

explained in Figure 5.1. Where i, j, k are number of neurons in the input, hidden and

output layers respectively.wij and wjk are synaptic weights of the input and hidden layers

respectively. f(.) is the activation function. Wi is the input values, Y1and Y2 are output

Hidden layers Input layers

Output layers wij wjk

Y1

f (.)

f (.)

f (.)

f (.)

f (.)

W1

W2

Wi

Y2

Figure 5.1: Simplified NN model with feed forward networks



88

values. In feed forward neural networks (FFNNs), the computation of the network is done

only in the forward direction. Synaptic weights are assigned to the inputs to the neuron

which can influence the abilities to take decision of the neural network model. The inputs

to the neuron are named as weighted inputs. The weighted inputs are gathered in the

hidden layers. If the summation of the weighted inputs goes beyond the predetermined

threshold value, the neuron electrifies, otherwise does not electrify. The scope of the

activation function is to restrict the output of the neuron amplitude.

5.2.2 Recurrent neural networks (RNNs)

The RNNs are those kinds of neural networks which have single or multiple feedback

loops. Due to the connections of feedbacks loops to the network structure, the information

can be accumulated and used later. The use of RNNs is preferred over FFNNs due to its

dynamic memory, self-recurrent and redundancy. The simple architecture of an RNN

structure is presented in Figure 5.2.

Here W, Y = Values of the input and output layers respectively, Z-1

= The delay units from

the output to the context layers, w= Synaptic weight of the neuron. i, j, k and l= The total

number of units (neurons) in the input, hidden, output and context layers respectively.

In the RNN model, the context units are formulated due to the feedback connections. Due

to the feedback connections, the structure can form a closed loop. In the context units, the

information can be collected and employed later. The context units act as an additional

memory to the network. The recurrences in the recurrent networks permit the network to

retain information from the earlier period and used it later. The recurrences in the network

Input units Hidden units Output units

Context units

Figure 5.2: Architecture of a RNN model

Z-1

W Y

wij wjk

wlj



89

act as a dynamic memory to the network structure. The feedbacks in the RNN can be

either of a global or local type.

Locally recurrent networks- In the locally recurrent networks, the feedbacks are

surrounded by neuron models. In this kind of network, the feedbacks are neither connected

to the neurons of progressive layers nor horizontal connections between the neurons of the

same layer. The structure of the locally recurrent neural networks is alike to static feed

forward structure, but the structure consists of dynamic memory neuron models.

Globally recurrent networks- In these types of networks, the feedbacks are either

connected between neurons of different layers or same layer. These networks include a

static multilayer perceptron. The networks also allow the non-linear mapping potentials of

the multilayer perceptron. The globally recurrent networks are classified into three types

i.e. fully recurrent and partially recurrent networks. The partially recurrent networks have

advantages over the fully recurrent networks that these recurrent connections are well

structured. These networks lead to greater training processes and less stability. The

numbers of units are strongly linked to the number of hidden or output neurons which

effectively control their flexibility. So additional recurrent links named as context units are

provided from the hidden or output units. Based on the additional units, the recurrent

networks are broadly classified into three types i.e. Jordan’s recurrent neural networks

(JRNNs), Elman’s recurrent neural networks (ERNNs), and Hopfield’s recurrent neural

networks (HRNNs).

Jordan’s recurrent neural networks (JRNNs)-

The simple architecture of a JRNNs model is presented in Figure 5.3. The architecture of

JRNNs was developed by Michel I. Jordan. Jordan [230] has added recurrent connections

from the network’s output to form context units. In Jordan networks, the context units are

also consigned to as the position layer. The outputs linked with every state are fed back to

the context units and merged with the inputs by characterizing the next state on the nodes

of the input. The entire state now comprises a new step for progression at the subsequently

time step. After numerous steps of procedures, the model presents on the context units,

along with the input units, is representative of the particular progression of the state that

the network has performed.



90

Elman’s recurrent neural networks (ERNNs)-

The ERNN is a partial recurrent neural network that recognizes patterns from the series of

values employing the back propagation method through time learning mechanism. Elman

[230] first developed the architecture in 1990. The simple architecture of an ERNN model

is shown in Figure 5.4. In ERNN, the recurrent links are given from the hidden layer to

the addition layer, which is known as context layer. The information from the hidden layer

is stored in the additional or context layer and again accumulated to the hidden layer. The

values from the preceding steps can be gathered and reused in the present time steps. The

Figure 5.3: Simple Architecture of JRNN model

Input layers Hidden layers Output layers

Context layers

Z-1

Z-1

Y1

Yn

W1

Wn

wij wjk

wlj

Figure 5.4: Simple architecture of ERNN model

Input layers Hidden layers Output layers

Context layers

Z-1

Z-1

Y1

Y2

W1

W2

wij wjk

wlj



91

accumulated information can be utilized later and this allows sequential and spatial pattern

recognition mechanism for the ERNN model.

Hopfield’s recurrent neural networks (HRNNs)-

A Hopfield is one type of RNN which was developed by Hopfield in 1982 [265]. The

model of simple HRNNs is shown in Figure 5.5. The HRNNs provide a context-

addressable memory structures with bidirectional threshold outputs. The HRNNs store

information both from psychology and neurology and develop a human memory model

that is known as associative memory. The HRNNs is a neural network which has

recognition of associative memory. The HRNNs are guaranteed to congregate a local

minimum but sometimes a wrong one. The HRNN structure provides an NN model for

recognizing the human memory.

The present Chapter is focused on knowledge-based studies on Jordan’s, Elman’s, and the

hybrid structure of both the Jordan’s and Elman’s recurrent neural networks. The results

obtained from the analyses of all network models are also compared with each other. The

knowledge-based studies are usually based on proficient and qualitative analysis. These

studies comprise the rule-based method. The rule-based methods, where the investigative

w11

w12 w13

w14 w21

w22 w23

w24 w31

w32 w33

w34

w41

w42

w43

w44

Y1 Y2 Y3 Y4

W1 W2 W3 W4

Figure 5.5: Simple architecture of HRNN model



92

rules can be invented from the structural process, unit function and simulation based

qualitative approach. In the present analysis, the faults are frequently identified by

reasoning some training symptoms with forward and backward directions along with the

propagating path of the neurons.

5.3 Use of Levenberg-Marquardt back propagation method

for RNN

The Levenberg-Marquardt (L.M) was developed by K. Levenberg and D. Marquardt [265]

independently. The advantages of the L.M back propagation algorithm over other

algorithms are that it is stable and fast. The L.M algorithm presents a computational

solution to a problem to minimize the non-linear function. This method is fast and has

steady convergence. The L.M algorithm is, in fact, the blend of two minimization process

i.e. the steepest descent method and Gauss-Newton method. The method comes from the

speed improvement of the Gauss-Newton technique and steadiness of steepest descent

technique. The fundamental concept of the L.M mechanism is that it executes an

interactive training process around the region with complex curvature. The L.M algorithm

changes to the steepest descent method to formulate a quadratic approximation and then

turned into the Gauss-Newton method to accelerate the convergence of the algorithm

during the training process.

According to, Yu and Wilamoski [266], the equation of the Levenberg-Marquardt back

propagation algorithm is given by-

1

1 ( ) T

k k k k k kw w J J X J e (5.1)

Where ‘ J ’ (the Jacobian matrix) has been calculated from the Gauss-Newton method. ‘X’

is the identity matrix. ‘ ’ is the combination coefficient. If the value of ‘ ’ approximates

to zero, the equation (5.1) will behave like Gauss-Newton method and, if the value of ‘ ’

is very large, then it will proceed as steepest descent method.

= (1/ ν), ‘ν’ is the step size or training constant.

e=Error vector= desired actual .

Where desired is the required output vector, actual is the actual output vector.



93

ε= Error function= 2

1

2 all training all outputspatterns

e (5.2)

The implementation of the L.M algorithm mainly depends on the calculation of ‘J’ and

organization of the training procedure iteratively for weight updating. In the L.M back

propagation method, the back propagation process is repeated for each output so as to

attain the successive rows of the Jacobin matrix. The error back propagating units is also

determined for each neuron (hidden and output) separately in L.M algorithm both for

forward and backward computation. Once the calculation of the Jacobian matrix is over,

then the next step is to arrange the training procedure of the network.

5.3.1 Steps for the organization of the training procedure using L.M

algorithm

Step 1- Generate the initial weight (wk)

Step 2- Determine the error (εk)

Step 3- Calculate the Jacobian matrix (J)

Step 4- As per equation (5.1); update the values to adjust the weight

1

1 ( ) T

k k k k k kw w J J X J e

Step 5- Calculate the total error (εk+1) with the modified weight

Step 6- If εk+1 > εk, Eliminate the step and improve the value of ‘ ’ with some factor and

repeat step-4 to update the value.

Step 7- If εk+1 ≤ εk, Allow the step and decrease the value of ‘ ’ with some factor and

repeat step as in step-5.

Step 8- Go to step 4 with the modified weight till calculated error is lesser than the

requisite error.

Like these procedures, the L.M algorithm is updated. The Jacobian matrix is computed,

and training processes are designed. As per the update rule, if the computed error becomes

smaller than the last error, it means that the quadratic approximation on the total error is

functioning, and the value of ‘ ’ should be decreased to minimize the significance of

gradient descent section. On the other hand, if the computed error is greater than the last



94

error, it is required to estimate a requisite curvature for proper quadratic approximation

and the value of ‘ ’ is to be increased.

5.4 Application of rule-based modified JRNNs for damage

identification in structure under moving mass

In the present analysis, rule-based JRNNs have been proposed for damage identification in

structures under moving mass and the architecture of the developed model has been shown

in Figure 5.6. The network consists of one input, output and context layers, and three

hidden layers. There is a main feedback connection from the output layer to the context

layer. The proposed network architecture is slightly modified to the original JRNN

network by introducing self-recurrent links in the output and context units. Thus, the

dynamic memory is provided employing feedback and self-recurrent links to the network.

The nodes in the context layer receive information through the feedback links from the

output layer, and outputs of nodes in the context layer provide information to the initial

hidden layer. Each node in the context, as well as, output layers has self-recurrent links to

itself. The self recurrent and the feedback links have one-time delay unit. The numbers of

units in the context unit are same as those of output units. It’s because the output values

can be exactly copied to the context units due to the feedback links. All the feed forward,

feedback and self-recurrent links of the proposed network are adapted by the Levenberg-

Marquardt algorithm which has been presented in equation (5.1). The proposed JRNN

model for damage identification in structure has been trained with 600 patterns of data for

attributing different conditions of the each of the structural system.



95

The adopted activation function for the hidden and context layers is ‘tan-sigmoid’ and that

of output the layer is ‘purelin’ functions respectively. The symbols used for the activation

functions in the hidden and output layers are f (.) and g (.) respectively. The self-recurrent

links also have one-time unit delay. During the training procedures and operation of the

network, the training patterns fed forward to the network encompass the following

components:

Figure 5.6: Architecture of modified JRNN model

Input layer

First hidden

layer

Output layer

Context layer

RD-1

v (m/s)

M (kg)

rcl1

rcd1

rcl2

rcd2

rcl3

rcd3

Z-1

Third hidden

layer

Second hidden

layer

(15 neurons) (15 neurons) (15 neurons)

(6 neurons)

(6 neurons)

(6 neurons)

RD-3

RD-2

RD-4



96

t=Total travelling time of the traversing mass to cross the beam

RD= Relative deflection= Deflection of cracked beam to uncracked beam at a particular

instant of time.

RD-1 =Relative deflections of the beam at time‘t/4’.

RD-2 = Relative deflections of the beam at time‘t/2’

RD-3= Relative deflections of the beam at time ‘3t/4’.

RD-4= Relative deflections of the beam at time‘t’.

W1= RD-1. W2= RD-2. W3= RD-3. W4= RD-4

W5= Speed of the moving mass (v).

W6= Magnitude of the moving mass (M).

1 = Relative first crack location (rcl1).

2 = Relative first crack depth (rcd1).

3 = Relative second crack location (rcl2).

4 = Relative second crack depth (rcd2).

5 = Relative third crack location (rcl3).

6 = Relative third crack depth (rcd3).

Where, i= 1, 2...N1, ‘N1’ is the total input nodes. r= 1, 2...N2, ‘N2’ is the total context

nodes. k=1, 2...O, ‘O’ is the total output nodes.

‘j1= j2 = j3=1, 2... S’, ‘S’ is the total number of neurons (nodes) in each of the first, second

and third hidden layers respectively (constant for all the nodes).

‘β’ is the self-recurrent value of the each node in the context and output layers respectively

which lie between 0 to 1.

Z-1

is the unit delay.

U1-6= The values of context units in the context layer.



97

1t

r

and t

r are the output values of the context node ‘r’ at time index ‘t-1’ and ‘t’

respectively.

t

j is the output values of the hidden node ‘j’ at time index ‘t’.

1t

k

and t

k are the output values of the output nodes at time index ‘t-1’ and ‘t’

respectively.

‘t-1’ is the time index which is delayed by one-time step due to the feedback links.

‘w’ is the weight of connection.

From the analysis of the network model (Figure 5.1), it has been observed that

1 1t t t

r k r (5.3)

The net input to the first hidden layer, 1 , 1 , 1

1 1

t tN Rt

j i i j r r j

i r

W w w

(5.4)

The net input to the second hidden layer, 2 1 1, 2

1

tSt

j j j j

j s

w

(5.5)

The net input to the third hidden layer or network is given by the following relation-

3 2 2, 3

2 1

tSt t

j j j j j

j

net w

(5.6)

( )t t

j jf net (5.7)

1

3 3,

3 1

tSt t

k j j k k

j

net w

(5.8)

( )t t

k kg net (5.9)

The numbers of nodes in each of the input, output and context layers are 6 and 15 in each

hidden layer. The number of neurons in each hidden layers are preferred by iterating

procedures of the training process and found to be 15 as suitable one. The approximation

error (ε) in the output nodes can be minimized using the updated weight factors relation,

i.e., new oldw w w , where ‘ ’, the learning rate is varying from 0 to 1. The

Levenberg-Marquardt algorithm has been applied to the proposed network to estimate the



98

crack locations and depth of the structural systems. The sum square error function is

characterized to evaluate the training procedures.

5.5 Application of rule-based modified ERNNs for damage

detection in structure subjected to moving mass

The Elman neural network, a partial recurrent neural network was first recommended by

Elman [230]. The network designed by Elman lies between the conceptions of feed

forward and recurrent network. Like the Jordan networks, the ERNNs have four layers i.e.

input, hidden, output and context layers. The context layer is formed by the feedback links

from the hidden layer. From the context layer, the dynamic memory is provided to the

network. In this analysis, modified ERNNs are proposed for the damage identification in

the structure under moving load. The architecture of the modified ERNNs model is

presented in Figure 5.7. The proposed architecture consists of one input, one output, three

hidden and two context layers. The numbers of neurons in the input and output layers are

6, and those of in each hidden and context layer are 15. The context layer-1 gets

information from the initially hidden layer through feedback links and supplies the outputs

to the primary hidden layer. The context layer-2 receives feedback signals from the

context layer-1 and the outputs of the nodes of the context layer-2 are fed forward to the

first hidden layer. Thus, the context layers-1 and 2, provide dynamic memories to the

network using feedback connections. As the networks have multi-hidden layers, the

feedback connections are provided from the nodes in a hidden layer to the nodes in the

corresponding previous hidden layer. All the nodes have self-recurrent links except those

in the input and output layers. The self-recurrent links in the nodes of the hidden layers

provide more generalizations to the network structure for identification of non-linear

systems.



99

Where, ‘i=1, 2...N’, ‘N’ is the total number of input nodes. ‘j1= j2= j3=1, 2,..S’, ‘S’ is the

total number of nodes in each of the hidden layer. ‘l1, =l2=1, 2,..T’, ‘T’ is the total number

of nodes in each of the context layer-1and 2.

‘k=1, 2,..O’, ‘O’ is the total number of nodes in the output layer.

‘β’ is the self-recurrent link value in the each node of the context layer-1,context layer-2,

first hidden, second hidden and third hidden layers respectively.

rcd1

rcl2

rcd2

rcl3

rcd3

rcl1

Figure 5.7: Architecture of modified ERNN model

Input layer

First hidden

layer

Output layer

Context layer-1

RD-1

v (m/s)

M (kg)

Z-1

Third hidden

layer

Second hidden

layer

Z-1

Z-1

Context layer-2

Z-1

(6 neurons)

(6 neurons)


(15 neurons) (15 neurons)

RD-2

RD-3

RD-4



100

X1-6 and V1-6 are the values of context nodes in the context layer-1 and 2 respectively.

1

1

t

l and 1

t

l are the net output of the nodes of the context layer-1 at time index ‘t-1’ and

‘t’ respectively.

1

2

t

l and 2

t

l are net the output of the nodes of the context layer-2 at time index ‘t-1’ and


1

1

t

j and 1

t

j are the net output values of the first hidden layer at time index ‘t-1’ and ‘t’

respectively.

1

2

t

j and 2

t

j are the net output values of the second hidden layer at time index ‘t-1’ and


1

3

t

j and 3

t

j are the net output values of the third hidden layer at time index ‘t-1’ and ‘t’

respectively.

1t

k

and t

k are the net output values of the output nodes at time index ‘t-1’ and ‘t’

respectively.

The other symbols used in the network have the usual meaning as those in JRNNs model.

From the analysis of the ERNNs model (Figure 5.7), it has been obtained that-

1 1

1 1 1

t t t

l j l (5.10)

1 1

2 1 2

t t t

l l l (5.11)

The net input to the first hidden layer is given by using the following relation-

1 1

1 , 1 1 2 1 2

1

Nt t t t t

j i i j j j l l

i

W w

(5.12)

The net input to the second hidden layer, 1 1

2 1 1, 2 2 3

1 1

St t t t

j j j j j j

j

w

(5.13)

The net input to the third hidden layer or to the network model is given by-

1

3 2 2, 3 3

2 1

St t t

j j j j j

j

w

(5.14)

The netjt = 3

t

j =t

j =f (netjt) (5.15)



101

The netkt = 3 3,

3 1

St

j j k

j

w

(5.16)

The net output of the proposed network is given by ( )t t

k kg net (5.17)

The numbers of nodes in each of the input, output layers are 6. 15 neurons are there in

each of the hidden and context layers. The ‘tan-sigmoid’ activation function is employed

in the hidden and context layer, while that in the output layer is ‘purelin’. The proposed

ERNNs have been trained using the Levenberg-Marquardt algorithm to find out the

locations and depth of the cracks on the structure. The ERNNs employs the approximation

error function (ε) in the output nodes to minimize the error value using the updated weight

factors relation, i.e., new oldw w w , where ‘ ’, the learning rate is varying from 0 to 1.

The sum square error function is implemented to evaluate the training process. Like the

JRNNs training process, the same training procedures are carried for the ERNNs model.

5.6 Application of rule-based modified hybridized JRNNs

and ERNNs for multiple damage detection in structure

subjected to moving mass

A novel recurrent neural network structure with the hybridization of both the modified

JRNNs and ERNNs has been presented in Figure 5.8. The hybrid structure is designed by

considering the architectural issue only. The designed structure consists of one input, one

output, three hidden and three context layers. The context layer-1, context layer-2, and

context layer-3 are formed due to the feedback connection from the first hidden layer,

context layer-1 and output layer respectively. Except the nodes in the input and output

layers, all the nodes in the network have self-recurrent connections. The feedback

connections in the hidden layers are provided from the nodes in a hidden layer to the

nodes in the previous hidden layer. Due to the self-recurrent and feedback links, the

dynamic memories are provided to the network for generalization of non-linear systems.

The self-recurrent and feedback links have one time-delay unit each. The hidden layer-1

receives signals from input and context layers-1, 2, 3, and feed forwards it to the hidden

layer-2.



102

Figure 5.8: Hybridized architecture of modified JRNN and ERNN

models

Input layer

First hidden

layer

Output layer

Context layer-1

RD-1

v (m/s)

M (kg)

Third hidden

layer

Second hidden

layer

rcd1

rcl2

rcd2

rcl3

rcd3

rcl1

Z-1

Z-1

Context layer-2

Z-1

Z-1

Context layer-3

Z-1

(6 neurons)

(15 neurons) (15 neurons)


(6 neurons)

(6 neurons)

RD-2

RD-3

RD-4



103

The nodes in the output layer produce the desired outputs by receiving information

through feed- forward links from the last hidden layer and due to the self-recurrent links.

1

3

t

l and 3

t

l are net the output of the nodes of the context layer-3 at time index ‘t-1’ and


The other symbols used in the network have the usual meaning as those in JRNNs and

ERNNs model. From the hybridization of the JRNNs and ERNNs model (Figure 5.8),

1 1

1 1 1

t t t

l j l (5.18)

1 1

2 1 2

t t t

l l l (5.19)

1 1

3 3

t t t

l k l (5.20)

The net input to the first hidden layer is given by using the following relation-

1 1

1 , 1 1 2 1 2 3

1

Nt t t t t t

j i i j j j l l l

i

W w

(5.21)

The net input to the second hidden layer, 1 1

2 1 1, 2 2 3

1 1

St t t t

j j j j j j

j

w

(5.22)

The net input to the third hidden layer or to the network model is given by-

1

3 2 2, 3 3

2 1

St t t

j j j j j

j

w

(5.23)

The netjt = 3

t

j = t

j =f (netjt) (5.24)

The netkt = 1

3 3,

3 1

St t

j j k k

j

w

(5.25)

The net output of the proposed network is given by- ( )t t

k kg net (5.26)

The numbers of nodes in each of the input, output and context layer-3 are 6. 15 neurons

are present in each of the hidden and context-1 and 2 layers. The activation function in the

hidden and context layers is ‘tan-sigmoid’, while that in the output layer is ‘purelin’. The

proposed hybrid structure of the modified JRNNs and ERNNs has been trained using the

Levenberg-Marquardt algorithm to predict the locations and depth of the cracks on the

structure. The approximation error (ε) function has been applied in the nodes of the output

to reduce the error values using the updated weight factors relation, i.e., new oldw w w ,



104

where ‘ ’, the learning rate, is varying from 0 to 1. The training processes have been

evaluated using the sum square error function. The training procedures are carried out in

the same manner to those of the JRNNs and ERNNs.

Table 5.1: Test patterns to the RNN model for cracked cantilever beam

Input data to the RNN model Output from the RNN model

RD-1 RD-2 RD-3 RD-4 v (m/s) M (kg) rcl-1 rcl-2 rcl-3 rcd-1 rcd-2 rcd-3

1 1.11 1.21 1.25 6.5 1.5 0.42 0.62 0.72 0.42 0.52 0.32

1 1.06 1.07 1.09 8.5 2.5 0.42 0.62 0.72 0.24 0.25 0.23

1.11 1.16 1.24 1.32 5.9 1.8 0.23 0.57 0.87 0.37 0.43 0.53

1.08 1.13 1.19 1.27 6.3 2.3 0.23 0.57 0.87 0.3 0.4 0.5

0.998 1.23 1.2 1.17 7.3 1.7 0.28 0.48 0.67 0.2 0.3 0.28

1 1.27 1.28 1.21 9.5 2.7 0.28 0.48 0.67 0.22 0.33 0.44

0.996 1.15 1.21 1.19 10 3 0.32 0.53 0.76 0.44 0.33 0.22

1 1.13 1.19 1.17 12 2.8 0.32 0.53 0.76 0.4 0.3 0.2

1 1.12 1.22 1.24 9 2.4 0.44 0.58 0.78 0.44 0.55 0.35

0.997 1.11 1.19 1.21 8.7 2.7 0.44 0.58 0.78 0.4 0.5 0.3

0 100 200 300 400 500 600 700 800 900 10004

4.5

5

5.5

6

6.5

7

7.5

Number of iterations

Su

m s

qu

ared

err

ors

JRNNs

ERNNs

Hybridisation of JRNNs and ERNNs

Figure 5.9: Plot of graph of iterations vs. sum square errors for RNNs methods

model



105

Tab

le 5

.2(a

): C

om

par

ison

of

resu

lts

bet

wee

n e

xper

imen

ts a

nd

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tion

of

rela

tiv

e cr

ack l

oca

tion

s (c

anti

lev

er b

eam

)

Exp

erim

enta

l

JRN

Ns

ER

NN

s

Hyb

rid

str

uct

ure

of

JRN

Ns

and

ER

NN

s

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

0.2

49

0.4

502

0

.64

81

0.2

319

0.4

207

0.6

091

0.2

353

0

.42

42

0.6

141

0.2

387

0.4

32

0.6

206

0.4

991

0.6

497

0

.84

81

0.4

643

0.6

087

0.7

908

0.4

714

0

.61

41

0.8

068

0.4

776

0.6

205

0.8

108

0.1

496

0.3

978

0

.62

95

0.1

396

0.3

727

0.5

875

0.1

407

0

.37

6

0.5

908

0.1

428

0.3

801

0.6

006

0.1

981

0.4

685

0

.69

85

0.1

866

0.4

391

0.6

534

0.1

887

0

.44

29

0.6

627

0.1

891

0.4

477

0.6

661

0.2

987

0.5

206

0

.60

03

0.2

808

0.4

853

0.5

596

0.2

809

0

.49

15

0.5

667

0.2

851

0.4

967

0.5

735

0.3

493

0.5

481

0

.71

95

0.3

234

0.5

117

0.6

701

0.3

285

0

.52

01

0.6

82

0

.33

34

0.5

235

0.6

879

0.4

491

0.5

991

0

.74

87

0.4

203

0.5

607

0.6

999

0.4

243

0

.56

61

0.7

047

0.4

281

0.5

714

0.7

145

0.3

982

0.4

994

0

.59

85

0.3

746

0.4

652

0.5

608

0.3

766

0

.47

29

0.5

631

0.3

805

0.4

777

0.5

713

0.2

201

0.6

192

0

.79

88

0.2

047

0.5

784

0.7

487

0.2

074

0

.58

81

0.7

575

0.2

101

0.5

913

0.7

627

0.3

20

0.5

583

0

.85

83

0.2

98

0

.52

44

0.8

072

0.3

012

0

.53

07

0.8

121

0.3

053

0.5

328

0.8

189

Aver

age

per

centa

ge

of

erro

rs

6.5

9

6.4

7

6.4

5

5.6

2

5.3

6

5.4

5

4.4

4

4.4

5

4.5

1

To

tal

per

centa

ge

of

erro

r 6

.5

5.4

7

4.4

3



106

Tab

le 5

.2 (

b):

Co

mp

aris

on

of

resu

lts

bet

wee

n e

xper

imen

ts a

nd

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tion

of

rela

tiv

e cr

ack d

epth

(ca

nti

lev

er b

eam

)

Exp

erim

enta

l

JRN

Ns

ER

NN

s

Hyb

rid

str

uct

ure

of

JRN

Ns

and

ER

NN

s

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

0.5

998

0.2

499

0.4

501

0.5

532

0.2

32

0.4

187

0.5

652

0

.23

56

0.4

233

0

.57

51

0.2

378

0.4

314

0.3

006

0.5

498

0.3

991

0.2

786

0.5

123

0.3

718

0.2

841

0

.52

5

0.3

767

0

.28

69

0.5

266

0.3

82

0.1

497

0.4

005

0.5

705

0.1

409

0.3

741

0.5

303

0.1

412

0.3

796

0.5

378

0

.14

31

0.3

817

0.5

466

0.2

001

0.3

501

0.5

396

0.1

879

0.3

269

0.5

05

0.1

887

0

.32

8

0.5

093

0

.19

11

0.3

342

0.5

167

0.2

496

0.2

996

0.4

49

0.2

327

0.2

796

0.4

243

0.2

347

0

.28

15

0.4

254

0

.23

84

0.2

856

0.4

295

0.2

301

0.4

302

0.6

404

0.2

116

0.4

03

0.5

998

0.2

161

0

.40

73

0.6

048

0

.22

01

0.4

112

0.6

135

0.3

193

0.2

698

0.5

297

0.2

984

0.2

52

0.4

965

0.3

007

0

.25

6

0.4

996

0

.30

61

0.2

577

0.5

072

0.3

5

0.4

607

0.5

998

0.3

275

0.4

27

0.5

598

0.3

298

0

.43

45

0.5

657

0

.33

4

0.4

405

0.5

752

0.3

701

0.6

003

0.2

002

0.3

464

0.5

581

0.1

876

0.3

506

0

.56

54

0.1

892

0.3

526

0.5

753

0.1

902

0.5

50

0.5

80

0.5

50

0.5

139

0.5

416

0.5

148

0.5

19

0

.54

64

0.5

216

0

.52

75

0.5

563

0.5

257

Aver

age

per

centa

ge

of

erro

rs

6.5

5

6.5

3

6.4

7

5.4

7

5.5

2

5.5

7

4.4

4

.42

4.3

3

To

tal

per

centa

ge

of

erro

r 6

.51

5.5

2

4.3

8



107

Tab

le 5

.3 (

a):

Co

mp

aris

on

of

resu

lts

bet

wee

n e

xp

erim

ents

and

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tio

n o

f re

lati

ve

crac

k l

oca

tio

ns

(sim

ply

su

pp

ort

ed b

eam

)

Exp

erim

enta

l

JRN

Ns

ER

NN

s

Hyb

rid

str

uct

ure

of

JRN

Ns

and

ER

NN

s

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

0.1

071

0.2

142

0.3

214

0.0

996

0.1

995

0.2

99

0.1

008

0

.18

84

0.3

025

0.1

019

0

.20

42

0.3

061

0.1

428

0.2

857

0.3

571

0.1

331

0.2

669

0.3

327

0.1

346

0

.26

92

0.3

367

0.1

361

0

.27

31

0.3

404

0.2

142

0.4

285

0.6

428

0.2

01

0.4

02

0.6

017

0.2

026

0

.40

6

0.6

075

0.2

048

0

.41

04

0.6

138

0.2

142

0.5

714

0.8

571

0.1

997

0.5

352

0.8

039

0.2

029

0

.54

0.8

118

0.2

053

0

.54

76

0.8

199

0.2

5

0.6

071

0.8

928

0.2

335

0.5

675

0.8

381

0.2

361

0

.57

43

0.8

47

0.2

398

0

.58

24

0.8

568

0.1

214

0.4

071

0.5

50

0.1

135

0.3

798

0.5

144

0.1

145

0

.38

42

0.5

206

0.1

159

0

.38

85

0.5

264

0.2

001

0.4

428

0.6

857

0.1

868

0.4

129

0.6

418

0.1

891

0

.41

74

0.6

484

0.1

906

0

.42

23

0.6

559

0.2

357

0.4

142

0.8

142

0.2

205

0.3

859

0.7

646

0.2

23

0

.39

15

0.7

722

0.2

243

0

.39

51

0.7

807

0.2

857

0.5

002

0.7

143

0.2

677

0.4

655

0.6

701

0.2

709

0

.47

24

0.6

752

0.2

733

0

.47

81

0.6

845

0.1

786

0.3

571

0.5

714

0.1

67

0.3

334

0.5

339

0.1

691

0

.33

67

0.5

392

0.1

705

0

.34

18

0.5

465

Aver

age

per

centa

ge

of

erro

rs

6.5

4

6.6

2

6.4

1

5.4

6

5.5

4

5.4

4

4.4

8

4.3

8

4.5

4

To

tal

per

centa

ge

of

erro

r 6

.52

5.4

8

4.4

6



108

Tab

le 5

.3(b

): C

om

par

iso

n o

f re

sult

s b

etw

een

ex

per

imen

ts a

nd

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tio

n o

f re

lati

ve

crac

k d

epth

(sim

ply

su

pp

ort

ed b

eam

)

Exp

eri

menta

l

JRN

Ns

ER

NN

s

Hyb

rid

str

uctu

re o

f JR

NN

s an

d

ER

NN

s

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

0.1

495

0.2

997

0.4

495

0.1

395

0.2

798

0.4

19

0.1

41

3

0.2

821

0.4

264

0.1

426

0

.28

5

0.4

284

0.2

007

0.3

995

0.5

003

0.1

863

0.3

756

0.4

667

0.1

886

0

.37

69

0.4

735

0.1

906

0

.38

1

0.4

768

0.5

495

0.5

50

0.5

498

0.5

162

0.5

159

0.5

151

0.5

209

0

.52

13

0.5

22

0.5

27

0

.52

74

0.5

251

0.1

998

0.1

996

0.1

994

0.1

866

0.1

866

0.1

861

0.1

886

0

.18

88

0.1

882

0.1

912

0

.19

06

0.1

901

0.3

499

0.3

489

0.3

502

0.3

273

0.3

274

0.3

265

0.3

309

0

.33

03

0.3

302

0.3

339

0

.33

41

0.3

346

0.4

801

0.3

797

0.4

805

0.4

499

0.3

56

0.4

482

0.4

537

0

.35

83

0.4

544

0.4

591

0

.36

29

0.4

599

0.4

996

0.3

99

0.1

994

0.4

688

0.3

748

0.1

863

0.4

716

0

.37

74

0.1

882

0.4

767

0

.38

13

0.1

904

0.5

303

0.2

505

0.1

785

0.4

962

0.2

336

0.1

679

0.5

015

0

.23

54

0.1

697

0.5

059

0

.23

75

0.1

716

0.3

50

0.4

50

0.5

485

0.3

271

0.4

214

0.5

167

0.3

303

0

.42

67

0.5

216

0.3

343

0

.42

88

0.5

268

0.1

994

0.2

997

0.4

006

0.1

865

0.2

801

0.3

75

0.1

883

0

.28

36

0.3

79

0.1

905

0

.28

52

0.3

826

Avera

ge p

erc

enta

ge o

f err

ors

6

.48

6.3

4

6.4

6

5.5

4

5.5

1

5.3

1

4.5

1

4.5

9

4.4

1

To

tal

perc

enta

ge o

f err

or

6.4

2

5.4

5

4.5



109

Tab

le 5

.4(a

): C

om

par

iso

n o

f re

sult

s b

etw

een

ex

per

imen

ts a

nd

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tio

n o

f re

lati

ve

crac

k l

oca

tio

ns

(fix

ed-f

ixed

bea

m)

Exp

eri

menta

l

JR

NN

s

ER

NN

s

Hyb

rid

str

uctu

re o

f JR

NN

s an

d

ER

NN

s

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

0.1

071

0

.22

85

0.3

357

0.0

996

0.2

128

0.3

12

0.1

007

0

.21

48

0.3

154

0.1

018

0

.21

73

0.3

19

0.1

642

0

.30

71

0.4

142

0.1

531

0.2

863

0.3

857

0.1

547

0

.28

93

0.3

901

0.1

562

0

.29

27

0.3

941

0.2

0

.44

28

0.5

857

0.1

862

0.4

141

0.5

462

0.1

887

0

.41

77

0.5

524

0.1

912

0

.42

26

0.5

584

0.2

642

0

.55

71

0.6

857

0.2

468

0.5

217

0.6

408

0.2

495

0

.52

68

0.6

472

0.2

53

0

.53

22

0.6

549

0.3

428

0

.65

71

0.8

357

0.3

208

0.6

16

0.7

845

0.3

243

0

.62

18

0.7

925

0.3

272

0

.62

85

0.7

966

0.2

642

0

.47

85

0.6

928

0.2

474

0.4

475

0.6

495

0.2

502

0

.45

11

0.6

562

0.2

518

0

.45

81

0.6

625

0.3

928

0

.53

57

0.8

214

0.3

682

0.5

02

0.7

694

0.3

725

0

.50

58

0.7

766

0.3

752

0

.51

18

0.7

866

0.4

285

0

.57

14

0.7

142

0.4

022

0.5

36

0.6

681

0.4

066

0

.54

03

0.6

746

0.4

105

0

.54

75

0.6

844

0.5

0

.64

28

0.7

857

0.4

684

0.6

037

0.7

358

0.4

748

0

.60

98

0.7

419

0.4

798

0

.61

67

0.7

501

0.3

214

0

.57

14

0.8

928

0.3

009

0.5

342

0.8

386

0.3

039

0

.54

1

0.8

456

0.3

068

0

.54

58

0.8

548

Avera

ge p

erc

enta

ge o

f err

ors

6

.48

6.4

2

6.4

7

5.4

2

5.5

3

5.5

4

4.4

7

4.4

2

4.5

1

To

tal

perc

enta

ge o

f err

or

6.4

5

5.4

9

4.4

6



110

Tab

le 5

.4(b

): C

om

par

iso

n o

f re

sult

s b

etw

een

ex

per

imen

ts a

nd

dif

fere

nt

RN

Ns

met

ho

d f

or

pre

dic

tio

n o

f re

lati

ve

crac

k d

epth

(fi

xed

-fix

ed b

eam

)

Exp

erim

enta

l

JRN

Ns

ER

NN

s

Hyb

rid

str

uct

ure

of

JRN

Ns

and

ER

NN

s

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

rc

d-1

rc

d-2

rc

d-3

0.1

490

0.3

192

0.4

697

0.1

396

0.2

981

0.4

378

0.1

411

0

.30

1

0.4

42

0

.14

22

0.3

041

0

.44

69

0.2

192

0.4

197

0.5

688

0.2

053

0.3

918

0.5

316

0.2

073

0

.39

58

0.5

367

0

.20

92

0.3

995

0

.54

27

0.4

995

0.4

985

0.4

995

0.4

693

0.4

678

0.4

668

0.4

715

0

.47

07

0.4

714

0

.47

58

0.4

767

0

.47

66

0.3

008

0.2

996

0.2

993

0.2

801

0.2

804

0.2

803

0.2

835

0

.28

32

0.2

83

0

.28

57

0.2

863

0

.28

69

0.3

998

0.3

991

0.3

984

0.3

748

0.3

749

0.3

743

0.3

773

0

.37

90

0.3

784

0

.38

16

0.3

824

0

.38

2

0.2

506

0.3

50

0.4

485

0.2

332

0.3

27

0.4

214

0.2

353

0

.33

23

0.4

256

0

.23

91

0.3

351

0

.43

05

0.4

504

0.3

506

0.2

495

0.4

215

0.3

283

0.2

342

0.4

269

0

.33

08

0.2

371

0

.43

12

0.3

354

0

.23

93

0.3

989

0.4

009

0.3

994

0.3

75

0.3

746

0.3

747

0.3

791

0

.37

75

0.3

788

0.3

827

0.3

824

0

.38

32

0.2

50

0.2

988

0.3

505

0.2

336

0.2

808

0.3

271

0.2

365

0

.28

37

0.3

306

0

.23

94

0.2

872

0

.33

5

0.2

607

0.3

807

0.5

60

0.2

426

0.3

551

0.5

257

0.2

457

0

.35

93

0.5

278

0

.24

84

0.3

635

0

.53

69

Aver

age

per

centa

ge

of

erro

rs

6.4

6

6.4

1

6.3

4

5.5

1

5.4

6

5.4

2

4.5

7

4.4

1

4.2

9

To

tal

per

centa

ge

of

erro

r 6

.4

5.4

6

4.4

2



111

Tab

le 5

.5(a

): C

om

par

iso

n o

f re

sult

s am

on

g v

ario

us

met

hod

s fo

r pre

dic

tio

n o

f re

lati

ve

crac

k l

oca

tio

ns

(fix

ed-f

ixed

bea

m)

Exp

eri

menta

l

FE

A

Theo

ry

Hyb

rid

str

uctu

re o

f JR

NN

s an

d

ER

NN

s

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

rc

l-1

rc

l-2

rc

l-3

0.3

005

0.5

214

0.8

714

0.2

94

0.5

111

0.8

526

0.2

923

0

.50

69

0.8

489

0

.28

71

0.4

972

0.8

336

0.2

571

0.4

5

0.8

142

0.2

517

0.4

408

0.7

957

0.2

492

0

.43

66

0.7

899

0

.24

58

0.4

303

0.7

785

0.2

004

0.4

428

0.6

928

0.1

956

0.4

331

0.6

785

0.1

941

0

.43

1

0.6

731

0

.19

09

0.4

233

0.6

622

0.1

714

0.4

142

0.6

0

.16

77

0.4

036

0.5

859

0.1

668

0

.40

23

0.5

847

0.1

633

0.3

952

0.5

728

0.2

775

0.5

918

0.8

064

0.2

711

0.5

782

0.7

868

0.2

694

0

.57

7

0.7

84

0

.26

44

0.5

636

0.7

689

0.2

785

0.5

937

0.8

079

0.2

727

0.5

809

0.7

87

0.2

711

0

.57

87

0.7

881

0

.26

58

0.5

65

0.7

728

0.2

796

0.5

945

0.8

083

0.2

729

0.5

803

0.7

89

0.2

72

0

.57

88

0.7

865

0

.26

67

0.5

653

0.7

709

0.2

142

0.5

714

0.8

571

0.2

087

0.5

572

0.8

373

0.2

086

0

.55

66

0.8

337

0

.20

41

0.5

456

0.8

184

0.3

214

0.5

714

0.8

928

0.3

135

0.5

567

0.8

737

0.3

123

0

.55

57

0.8

693

0

.30

73

0.5

448

0.8

529

0.4

285

0.5

714

2

0.7

142

0.4

183

0.5

591

0.6

987

0.4

159

0

.55

55

0.6

941

0

.40

93

0.5

473

0.6

822

Avera

ge p

erc

enta

ge o

f err

ors

2

.24

2.2

7

2.2

8

2.7

7

2.6

9

2.7

4

.52

4.5

8

4.4

7

To

tal

perc

enta

ge o

f err

or

2.2

6

2.7

2

4.5

2



112

Tab

le 5

.5(b

): C

om

par

iso

n o

f re

sult

s am

on

g v

ario

us

met

hod

s fo

r pre

dic

tio

n o

f re

lati

ve

crac

k d

epth

(fi

xed

-fix

ed b

eam

)

Exp

eri

menta

l

FE

A

Theo

ry

Hyb

rid

str

uctu

re o

f JR

NN

s an

d

ER

NN

s

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

0.5

51

0.3

508

0

.15

1

0.5

402

0.3

441

0.1

482

0.5

368

0

.34

21

0.1

472

0

.52

6

0.3

357

0.1

444

0.2

01

0.4

21

0

.30

12

0.1

964

0.4

127

0.2

952

0.1

952

0

.40

99

0.2

938

0

.19

16

0.4

006

0.2

869

0.4

43

0.3

302

0

.22

08

0.4

332

0.3

231

0.2

16

0.4

308

0

.32

11

0.2

149

0.4

22

0.3

158

0.2

104

0.4

81

0.3

83

0

.28

11

0.4

697

0.3

742

0.2

748

0.4

683

0

.37

29

0.2

733

0

.45

76

0.3

649

0.2

682

0.2

091

0.3

006

0

.50

51

0.2

039

0.2

943

0.4

946

0.2

039

0

.29

2

0.4

905

0

.19

99

0.2

869

0.4

838

0.1

72

0.3

81

0

.46

08

0.1

684

0.3

72

0.4

499

0.1

672

0

.36

99

0.4

492

0

.16

43

0.3

632

0.4

415

0.4

143

0.2

106

0

.30

17

0.4

049

0.2

054

0.2

944

0.4

025

0

.20

43

0.2

936

0

.39

55

0.2

011

0.2

884

0.1

51

0.2

504

0

.35

04

0.1

474

0.2

444

0.3

417

0.1

471

0

.24

37

0.3

421

0

.14

44

0.2

395

0.3

347

0.3

042

0.4

14

0

.51

11

0.2

968

0.4

04

0.4

999

0.2

962

0

.40

35

0.4

971

0

.29

08

0.3

956

0.4

877

0.6

121

0.6

013

0

.60

1

0.5

989

0.5

877

0.5

892

0.5

964

0

.58

48

0.5

848

0

.58

51

0.5

756

0.5

731

Avera

ge p

erc

enta

ge o

f err

ors

2

.24

2.2

2

2.1

6

2.6

6

2.7

2

.61

4.5

2

4.4

8

4.4

7

To

tal

perc

enta

ge o

f err

or

2.2

2

.65

4.4

9



113

5.7 Discussion and Summary

The modified JRNNs, ERNNs and the hybrid structure of both the JRNNs and ERNNs

structures are studied in the present Chapter. The studies have been focused on the

architectural issues. The Levenberg-Marquardt algorithm has been applied to train all the

structures. 600 training patterns are generated for each of the damaged structure to train all

the network models. Some of the testing data for the damaged cantilever beam are

presented in Table 5.1. 1000 numbers of iterations are carried out to train each of the

network models. The relative locations and depth of cracks are determined in different

structures under transit mass with multiple cracks using the recurrent neural networks

methods. The JRNNs, ERNNs, and the hybridisation of both the JRNNs and ERNNs

recurrent neural analysis are applied to predict the locations and depth of cracks using the

Levenberg-Marquardt training algorithm.

The results obtained from the experimental analysis and different RNNs methods are also

presented in Tables 5.2- 4. A graph (Figure 5.9) has been plotted against the number of

iterations and the errors obtained from various RNNs methods to show the accuracies of

the proposed methods. From the analysis of Figure 5.9, it has been observed that the errors

are decreasing with the increase of the number of iterations. The errors in case of the

hybridisation of JRNNs and ERNNs method are less than those in JRNNs and ERNNs

methods. The comparison of results for relative crack locations and depth obtained from

the experiment and different RNNs methods are presented in Tables 5.2, 5.3 and, 5.4 for

cantilever beam, simply supported beam and fixed-fixed beam respectively.

The variation of results between the experiments and JRNNs, ERNNs, and the

hybridisation of both the JRNNs and ERNNs methods are near about with average errors

of 6.46%, 5.47% and 4.4% respectively. The variation of results among the experiments,

FEA and theoretical analyses for the case of fixed-fixed beam is presented in Tables 5.5

(a) and (b). The deviation of results between the experiment and FEA are of near about

2.2%, while those with theoretical analyses are of 2.7% respectively. The Levenberg-

Marquardt training algorithm [263] was applied to train each of the recurrent neural

networks models with 1000 number of iterations. From the analyses of results obtained

from the different recurrent neural network methods, it has been remarked that the

hybridization of the JRNNs and ERNNs yields better results as comparison to JRNNs and

ERNNs methods individually. Thus, the hybridised JRNNs and ERNNs perform better

and proximity results for damage detection in structure under transit mass.

114

Chapter 6

APPLICATION OF STATISTICS

BASED DAMAGE IDENTIFICATION

PROCEDURE FOR STRUCTURES

UNDER TRANSIT MASS

6.1 Introduction

The well-defined fault diagnosis procedures have become a significant problem in modern

mechanical control theory. Early detection of faults in structures provides a potential tool

to execute significant preventing actions. The key features of fault detection methods are

the rule based models which are used for decision making methods. During the normal

operation, the structures under transit mass are suffered from many changing ecological

and operational conditions which can affect the performance of the structural systems. If

the variances produced by the ecological and operational conditions are not appropriately

explained, then the sensitivity of the fault identification approach can be reduced. The

damage detection approaches based on statistical analyses plays a principal role in quality

control and improvement. Statistical analyses are the compilation of methods for preparing

decisions about a method which depend on the investigation of the data contained in a

given sample or population. The statistics based methods present the principal features by

means a product can be sampled, trained and monitored. The information in the sample

data is utilized to improve and control the training and monitoring process. In the present

Chapter, the damage detection procedure is in cast in the perspective of a statistical pattern

recognition problem. The proposed damaged detection method is developed using time

series analysis in the statistical process control (SPC) domain. The analysis has been

carried out using the IBM SPSS 20 software package.

Chapter 6 Application of Statistics Based Damage Identification Procedure for

Structures under Transit Mass

115

6.2 Overview of Statistical process control (SPC) method

Statistical process control (SPC), an influential collection of problem solving tools which

is useful to achieve process stability and improve capability through the reduction of

variability. The key features of SPC based variability reduction program is the continuous

improvement on timely programs. In the context of SPC, the problem can be described as

a four-part procedures namely operational evaluation, acquisition of data, feature

extraction and selection of data, and development of statistical model. The present study is

focused on the feature extraction and selection of data, and development of statistical

model. The development of statistical model is concerned with the execution of the

mechanism which can examine the distribution of extracted features to calculate the

damage configuration of the structure. The mechanism implemented in the statistical

model improvement basically categorized in three groups namely classification of group,

regression analysis and outlier identification. The present analysis falls on the regression

analysis category.

The principal role of the SPC technique is to identify the happening of the assignable

defects so that the process can be investigated and necessary preventive action may be

taken before the happening of any nonconforming units. Statistical process control (SPC)

technique can be applied to any procedures. The SPC method has seven important tools

known as ‘the magnificent seven’ namely-

(i) Check sheet. (ii) Steam and leaf plot. (iii) Cause and effect diagram.

(iv) Pareto chart. (v) Scatter diagram. (vi) Defect concentration diagram.

vii) Control chart.

The proposed method is focused on the control chart analysis.

6.3 Construction and analysis of control chart

The control chart analysis is an online monitoring process used for the prediction of

various parameters through proper information. The control chart acts like an important

tool to improve various process parameters. The control chart analysis is the graphical

representation of quality characteristics which has been calculated from a sample number

versus the sample characteristics. The control chart is a tool which can describe the



116

statistical process control in an exact and precise manner. The chart is applied for online

inspection of parameter estimation. The representation of a typical control chart is shown

in Figure 6.1.

The data collected from a given sample or population are used to prepare a control chart.

The centre line of the control chart represents the normal average estimated value of the

corresponding control state. If any point that lies outside of the control limit, then that

state is regarded as out of control. So survey and remedial actions are needed to find out

the faults and remove the causes responsible for this kind of behaviour. So control limits

are prepared. According to Montgomery [262], the control limits can be prepared

according to the following equations:

LCL= /2 ww az

s

(6.1)

UCL= /2 ww az

s

(6.2)

Where ‘ s ’ is the sample size, ‘w’ is the sample statistics, ‘µ’ is the mean vale and ‘σw’ is

the standard deviation. The value of ‘ w

s

’ can be tabulated from different sample or

population sizes [262].

The control chart uses the population average ( x ) to supervise the process mean. The

objectives of the control chart are to keep the process in control state. The implementation

of control chart in the SPC method is a significant step for early exclusion of assignable

Sample number

Sam

ple

ch

arac

teri

stic

s

Centre line

Upper control limit (UCL)

Lower control limit (LCL)

Figure 6.1: Architecture of control chart



117

causes. There are two types of control charts namely X-chart and R-chart. The current

analogy is dealt with X bar-chart for the control chart analysis.

6.4 Overview of Autoregressive model

It is required to develop a model which can be useful to determine the possibility of a

future value lying between two specific limits. The model may be named as probability or

stochastic model. So for the analyses of the times series, the construction of the model is

required. The present analysis is manifested on the analysis of Auto Regressive (AR)

model. A probability or stochastic model which can be very much useful in the

representation of a specific practically happening series is the Auto Regressive (AR)

model. In the AR model, the present value of the progression is usually represented as a

finite, linear aggregate of past values of the progression and a shock ‘at’. The shocks are

the unsystematic drawing from a linear distribution with zero mean and variance σw2. The

random variables of shocks ‘at, at-1, st-2, at-3.........’ are also known as white noise process.

The representation of a probability model in time series analysis is shown in Figure 6.2.

The probability models based on time series analysis are mostly dependent consecutive

values may be generated from a sequence of independent variables ‘at’ (shocks). The

shocks or white noise are randomly drawn from a fix distribution. These are assumed to be

normal with zero mean and constant variance. The shock or white noise procedure ‘at’ can

be transformed to other process ‘zt’ by a liner filter as in Figure 6.2. The liner filtering

procedures usually gets the summed values of the past shock. So

1 1 2 2 ... ( )t t t tz a a a b (6.3)

Where 2 3

1 2 3( ) 1 ... c

cb b b b b

Shock

or

White noise Outputs

Linear filter

Weights

Figure 6.2: Representation of a probability process as the outputs

from a liner filter



118

The representation of a probabilistic or stochastic process as the liner productivity from a

linear filter may be given the equation as:

1 1 2 2 3 3

1

....

or

t t t t c t c t

c

t j t j t

j

z z z z z a

z z a

(6.4)

Where ‘ 1 2 3, , ,.... c ’ are the parameters representing finite set of weights.

Where ‘t, t-1, t-2, ...’ are the uniformly spaced time over 1 2, , ,....t t t t cz z z z and

t tz z .

‘ tz ’ is the deviation of the process from the specified origin. The common normal

procedures as in equation (6.3), permits the process to represent ‘ tz ’ as a weighted sum of

present and past values of the shock or white noise process ‘at’. The present deviation ‘ tz ’

from the mean or level ‘µ’ is regressed on the preceding deviations 1 2, ,....t t t cz z z of the

proposed process.

The method explained by equation (6.3) is named as an AR (c) process with order ‘c’.

This is due to a liner model:

1 1 2 2 3 3 .... c cz x x x x e (6.5)

Here ‘z’ is the dependent parameter to the set of independent parameters, 1 2 3, , ,... cx x x x

and ‘e’ is the error term. The parameter ‘z’ is said to be regressed on the past outputs of it.

So the process is autoregressive. According to equation (6.3), the equation (6.4) can be

also expressed in the equivalent form i.e.

2

1 2(1 ... )c

c t tb b b z a (6.6)

The equation (6.6) can also be written as: ( ) t tb z a (6.7)

1( )t tz b a (6.8)

The AR (c) may be treated as the outcome ‘ tz ’ from the liner filter model having transfer

function ‘ 1( )b ’ with the input ‘ ta ’ to the model.

To avoid over parameterization, it is better to keep the order of the AR model at lower

level.



119

6.5 Application of Auto Regressive (AR) model based

method for damage detection in structures subjected to

traversing mass

The damage detection procedures for beam types structures subjected to traversing mass

are carried out using the AR model. For the AR model, 1000 patterns are generated for

each of the structures (cantilever, simply supported and fixed-fixed) at different

configurations of the moving mass structural systems. Out of 1000 patterns, 300 patterns

are for undamaged beam structures while 700 patterns are for damaged structures. The

time- displacement responses histories are determined for all the 1000 numbers of patterns

with 500 numbers of observations intervals. The mean, standard deviation and variances

are computed for each of the pattern. Then the average of the means and standard

deviations are computed by considering all the patterns. The data analysis for the control

chart are done using IBM SPSS software package. Initially, the control charts have been

prepared to know the existence of damages in the structures. The centre line (CL), upper

cutter limit (UCL) and lower control limit (LCL) are computed by considering the

undamaged structures. The detailed analysis of the control charts are represented in

Figures 6(a), 6(b) and 6(c) for the cantilever, simply supported and fixed-fixed beam

structures respectively. If the points lie above the UCL and below the LCL, then it shows

the possibilities of existence of damages in those structures. Selection of group size is an

important issue for preparing X bar control chart.

When the group size is more than one, then the mean of the data within one group is

selected as the chart variable and make one point in the subsequent chart. The selection of

group size has direct impact for the determination of the control limits and thus influences

the sensitivity of the control chart. The statistical process control (SPC) method

particularly based on two types of learning algorithm i.e. unsupervised and supervised.

The SPC method is applied to determine the relative crack locations and crack depth. In

unsupervised learning algorithm for damage detection in structures, the data from the

damaged states are not available, while in supervised learning algorithm, the data are

available.



120

Figure 6.3: Data analysis in SPSS windows

-0.5

0

0.5

1

1.5

2

2.5

1 5 9 13 17 21 25

Sam

ple

ob

serv

atio

ns

Sample numbers

sample measure

CL

UCL

LCL

Figure 6.4(a): Control chart for cantilever beam



121

In the present analysis, initially supervised learning algorithm is adopted to verify the

accuracy of the proposed method. In the later part, unsupervised learning algorithm is

implemented for fault detection in structure using the statistical process controller. The

autoregressive (AR) method is focused in this analysis.

In the present analogy, the extractions of data are carried out by theoretical-numerical

solution, FEA and experimental methods. The data compression method is used to convert

the time series data from multiple measurement points to single point.

Figure 6.4(b): Control chart for simply supported beam

-0.5

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30

Sam

ple

ob

serv

atio

ns

Sample numbers

sample observations

CL

UCL

LCL

-0.2

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

Sam

ple

ob

serv

atio

ns

Sample numbers

Sample observations

CL

UCL

LCL

Figure 6.4(c): Control chart for fixed-fixed beam



122

If ( )i jy t represents the response time histories at ‘n’ measurements points or positions and

sampled at ‘r’ time intervals, then a vector has been formed to define the response

components of subsequent ‘n’ measurement positions. The time history response at a

given time ‘ ( )jt ’ is expressed in the following way-

1 2 3( ) [ ( ) ( ) ( )... ( )]T

i j j j j n jy t y t y t y t y t (6.9)

Where the value of ‘i’ varies from 1 to n. ‘n’ is the number of locations where the

vibration based displacement-time history to be measured along the beam structures. The

displacement-time history data are not only correlated to each other but also to each

other’s previous data.

The equation (6.9) can also be written as:

1 2 3[ , , ... ]T

t t t t nty y y y y (6.10)

Then, the covariance matrix (Ω) of size n n among all the measurement positions over

the entire time intervals is given as: 1

( ) ( )r

T

j j

j

y t y t

(6.11)

Feature extraction is the method which can identify damage sensitive characteristics from

the measured dynamic response of structures that permits one to differentiate between

cracked and uncracked structures. It is difficult to distinguish the time series data from the

damaged and undamaged states by visual inspection. So other features of data extraction

are required for damage identification in structure. In the AR (c) model, the present value

in the time series analysis is represented as the linear combination of the past ‘c’ values.

The autoregressive model with order ‘c’, AR (c), can be expressed as:

1

c

t j t c t

j

y y a

(6.12)

Where ty is the history of time response at time ‘t’, ta is the random error or shock value

at zero mean and constant variance.

The equation (6.12) can also be written as in the form i.e.

1 1 2 2 3 3 ...t t t t c t c ty y y y y a (6.13)

Where, 1 2( , ,... ,... )j c , ‘ j ’ are the coefficient of AR (c) model which represents

the damage sensitive factors.

The values of ‘ j ’ are assessed by fitting the AR (c) model to the displacement- time

response data by applying the Yule-Walker method. [Box et al, 264]. The values of the



123

coefficient of the AR model have been calculated with a liner linear least squares

regression using the IMB SPSS software package. The order of the AR (c) model has been

calculated by verifying Gaussianity and randomness of the estimation errors by trial and

error approaches.

The ‘c’ matrices can be formed for each set of the displacement-time histories data which

is the constituent of ‘ j ’ ( n n matrix) and is represented below:

11j 12j 13j 1nj

21j 22j 23j 2nj

n1j n2j n3j nnj

ψ ψ ψ ...ψ

ψ ψ ψ ...ψ

ψ ψ ψ ...ψ

j

(6.14)

The vibration based displacement-time response of the undamaged and damaged

structures are compared indirectly to extract the damage sensitive factors. The damaged

sensitive factors ( j ) have been obtained by fitting the AR (c) model to the displacement-

time history data for each of the response data. The total undamaged data set are divided

into two sample groups namely reference data sample set (SampleR) and healthy data

sample set (Samplehd

), while those of cracked data set are named as Samplecd

. The

SampleR has been utilized to extract damage factors for all data set for future comparison.

The damage sensitive factors for Samplehd

and Samplecd

are also determined. The

extracted damage sensitive factors or coefficient of AR (c) model from the healthy and

damaged states are compared with the reference data sample set (SampleR).

Measuring the variation of damage sensitive factors which occur in the predicted

coefficient of AR model is due to the existence of cracks in the structures. The magnitudes

of variations in the coefficient of AR model or damage sensitive factors are determined

statistically (Fisher Criterion) with respect to the healthy state of the structure. The

coefficient of AR model provides the information about the vibration response data of the

structure. The amount of variation obtained in the coefficients of the AR model at

different locations of structures provides the information about the possible locations of

cracks in the beam structures.

The Fisher Criterion has been applied to calculate the actual variation of coefficient of AR

model or damage sensitive factors for both the damaged and healthy states. The Fisher

criterion is explained as follows:



124

2

cd hd

criterion

cd hd

FisherV V

(6.15)

Where ‘ and cd hd ’ are the respective means of damaged sensitive factors of cracked and

healthy states respectively. ‘ and cd hdV V ’ are the respective variances of damaged sensitive

factors of cracked and healthy states respectively. The value of the Fisher criterion will be

more at the possible damage locations. The greater values of the Fisher criterion provide

the information about the sudden increase in the response of the structures and thus able to

identify the damage locations.

Once the identification of the location of cracks is over, then it is desired to quantify the

severities of the identified cracks. The probability density function (PD) and fourth order

statistical moment method (FSMM) are introduced to quantify the severities of cracks

[Wang et al., 246]. The probability density function under Gaussian distribution process

can be articulated as: PD=2( ( )/2 )1

( )2

x xx

x

x e

(6.16)

Where ‘x’ is the amplitude or structural response of the beam structures, ‘ ’ is the

probability density function, ‘ and x x ’ are the mean and standard deviation of the

amplitude ‘x’.

The mathematical expression for the fourth order statistical moment for the amplitude or

structural response ‘x’ can be expressed as:

4 4

4 ( ) ( ) 3

s

x xx x dx (6.17)

The fourth order statistical moment for stiffness vector has been calculated for both the

undamaged and damaged structures. Incorporating the stiffness parameter with an initial

value to equation (4.1), Chapter-4, and the stiffness matrix and the displacement response

of the structures of the predicted crack locations are obtained using the finite element

modelling. The finite element model updating method (Newmark’s integration approach)

has been carried out to measure the cracks severities of the predicted crack element. The

displacement response of the cracked elements obtained from the finite element

modelling, from which the second order central difference approach has been applied to

get the strain values of the corresponding elements of the structures.



125

Let 4 ( )k is the simulated statistical moment for displacement response of the structure

with stiffness parameter ‘k’.

‘4

a ’ is the actual statistical moment for the measured displacement response of the

structure.

The residual vectors ( ( )R k ) between the simulated and actual displacement response of

the structure is given by the following equation:

4 4

4

( )( )

( )

akR k

k

(6.18)

The least square method has been applied to minimize the variation between the measured

and actual statistical moments. The information about the stiffness parameters of the

elements are obtained by the least square optimization method. From the element stiffness

parameter, the severities of the damages are quantified.

Table 6.1(a): Comparison of results among FEA, Theoretical and SPC methods for estimation of

relative crack locations (cantilever beam)

FEA Theoretical SPCM

rcl-1 rcl-2 rcl-3 rcl-1 rcl-2 rcl-3 rcl-1 rcl-2 rcl-3

0.182 0.3256 0.5025 0.1798 0.3214 0.4968 0.1784 0.3195 0.4937

0.2734 0.4214 0.5828 0.2705 0.4169 0.5758 0.2679 0.413 0.5731

0.342 0.485 0.605 0.3384 0.4806 0.5977 0.3346 0.4745 0.5936

0.375 0.506 0.628 0.3715 0.5002 0.6204 0.3665 0.4955 0.6139

0.512 0.624 0.705 0.507 0.6167 0.6961 0.5024 0.6112 0.6911

0.395 0.5525 0.715 0.3914 0.5463 0.7065 0.3857 0.5416 0.7018

0.407 0.576 0.727 0.4038 0.5697 0.7186 0.3989 0.5652 0.7139

0.428 0.526 0.735 0.4223 0.5201 0.7269 0.4195 0.5168 0.7208

0.456 0.639 0.743 0.4511 0.6313 0.7341 0.4462 0.6276 0.7284

0.578 0.676 0.825 0.5703 0.6683 0.8147 0.5649 0.6637 0.8116

Average percentage of error 1.06 1.12 1.19 2.1 1.96 1.85

Total percentage of error 1.12 1.97



126

Table 6.1(b): Comparison of results among FEA, Theoretical and SPC methods for estimation of

relative crack depth (cantilever beam)


rcd-1 rcd-2 rcd-3 rcd-1 rcd-2 rcd-3 rcd-1 rcd-2 rcd-3

0.157 0.262 0.282 0.1548 0.2591 0.2787 0.154 0.2568 0.2753

0.172 0.243 0.265 0.1694 0.2410 0.2614 0.1683 0.2385 0.2592

0.185 0.268 0.325 0.1831 0.2672 0.3215 0.1815 0.2624 0.3173

0.205 0.285 0.335 0.2031 0.2834 0.3306 0.2007 0.2790 0.3278

0.325 0.408 0.306 0.3222 0.4022 0.3037 0.3178 0.3991 0.2994

0.273 0.312 0.423 0.2704 0.3067 0.4192 0.2672 0.3056 0.4116

0.281 0.35 0.432 0.2785 0.3427 0.4273 0.2763 0.3432 0.4231

0.312 0.423 0.525 0.3076 0.4145 0.5195 0.3062 0.4150 0.5136

0.501 0.521 0.534 0.4953 0.5141 0.5249 0.4936 0.5121 0.5223

0.408 0.415 0.395 0.4046 0.4104 0.3903 0.4028 0.4082 0.3857



Table 6.2(a): Comparison of results among FEA, Theoretical and SPC methods for estimation of

relative crack locations (simply supported beam)



0.2178 0.3607 0.4321 0.2154 0.3571 0.4265 0.2142 0.3537 0.4227

0.2285 0.3714 0.4428 0.2259 0.3668 0.4376 0.2245 0.3631 0.4344

0.3042 0.45 0.5142 0.3019 0.4451 0.5068 0.2976 0.4408 0.5033

0.3142 0.3857 0.4571 0.3113 0.3813 0.4518 0.3074 0.3786 0.4486

0.3642 0.4642 0.5164 0.3597 0.4592 0.5106 0.3571 0.4555 0.5073

0.4785 0.5571 0.6021 0.4732 0.5505 0.5947 0.4703 0.5473 0.5914

0.55 0.6928 0.7285 0.5447 0.6846 0.7202 0.5404 0.6794 0.7141

0.5857 0.6857 0.7714 0.5806 0.6778 0.7627 0.5746 0.6727 0.7547

0.6285 0.75 0.8285 0.6211 0.7415 0.8192 0.6161 0.7381 0.8116

0.7014 0.8014 0.8857 0.6917 0.7916 0.8738 0.6866 0.7874 0.8671





127

Table 6.2(b): Comparison of results among FEA, Theoretical and SPC methods for estimation of

relative crack depth (simply supported beam)



0.173 0.265 0.325 0.1715 0.2624 0.3215 0.1696 0.2594 0.3186

0.328 0.402 0.505 0.323 0.3971 0.5001 0.3223 0.3932 0.4942

0.423 0.429 0.515 0.4179 0.4246 0.5094 0.4151 0.4209 0.5059

0.385 0.455 0.513 0.3815 0.4508 0.5087 0.3774 0.4439 0.5022

0.602 0.607 0.417 0.5969 0.6003 0.4140 0.5896 0.5953 0.4093

0.201 0.306 0.508 0.1987 0.3034 0.5047 0.1965 0.2991 0.4981

0.343 0.521 0.156 0.3395 0.5169 0.1532 0.3355 0.5097 0.1528

0.162 0.276 0.322 0.1601 0.2735 0.3182 0.1585 0.2695 0.3158

0.129 0.368 0.492 0.1274 0.3653 0.4858 0.1263 0.3613 0.4815

0.398 0.565 0.602 0.3932 0.5593 0.5964 0.3907 0.5531 0.5915





128

Ex

per

imen

tal

Theo

reti

cal

F

EA

S

PC

M

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

rcl-

1

rcl-

2

rcl-

3

0.2

017

0.3

264

0.3

821

0.1

964

0.3

178

0.3

821

0.1

975

0.3

183

0

.37

38

0.1

938

0.3

146

0.3

668

0.2

281

0.3

013

0.3

642

0.2

214

0.2

928

0.3

642

0.2

246

0.2

946

0

.35

55

0.2

193

0.2

903

0.3

504

0.3

165

0.4

716

0.5

428

0.3

071

0.4

571

0.5

428

0.3

092

0.4

589

0

.52

92

0.3

041

0.4

537

0.5

202

0.3

834

0.4

936

0.5

928

0.3

714

0.4

785

0.5

928

0.3

746

0.4

817

0

.57

88

0.3

678

0.4

749

0.5

683

0.4

183

0.5

363

0.6

214

2

0.4

071

0.5

214

0.6

214

2

0.4

084

0.5

239

0

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71

0.4

023

0.5

155

0.5

959

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849

0.6

304

0.6

928

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0.6

142

0.6

928

0.4

726

0.6

148

0

.67

61

0.4

673

0.6

046

0.6

634

0.5

516

0.7

219

0.8

428

0.5

357

0.7

014

0.8

428

0.5

381

0.7

043

0

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56

0.5

314

0.6

944

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0.6

329

0.7

513

0.8

928

0.6

142

0.7

285

0.8

928

0.6

168

0.7

334

0

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26

0.6

048

0.7

229

0.8

563

0.6

780

0.7

926

0.8

071

0.6

571

0.7

714

0.8

071

0.6

623

0.7

757

0

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84

0.6

489

0.7

633

0.7

739

0.1

910

0.4

979

0.6

714

0.1

857

0.4

857

0.6

714

0.1

864

0.4

858

0

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13

0.1

833

0.4

778

0.6

437

Aver

age

per

centa

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of

erro

r 2

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2.7

8

2.6

7

2.2

8

2.4

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3.9

6

3.8

1

4.1

To

tal

per

centa

ge

of

erro

r 2

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2.3

5

3.9

5

Tab

le 6

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): C

om

par

ison o

f re

sult

s am

ong E

xper

imen

tal,

FE

A,

Th

eore

tica

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d S

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met

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for

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(fix

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129

Ex

per

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tal

Theo

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F

EA

S

PC

M

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

rcd

-1

rcd

-2

rcd

-3

0.2

696

0

.32

38

0.4

296

0

.26

2

0.3

14

0

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6

0.2

638

0.3

162

0

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08

0.2

589

0.3

112

0.4

133

0.3

528

0

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23

0.5

112

0

.34

3

0.5

17

0

.49

6

0.3

439

0.5

193

0

.50

07

0.3

385

0.5

118

0.4

921

0.3

260

0

.43

44

0.5

281

0

.31

7

0.4

22

0

.51

3

0.3

186

0.4

235

0

.51

75

0.3

127

0.4

179

0.5

081

0.5

394

0

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18

0.1

213

0

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4

0.4

29

0

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8

0.5

264

0.4

313

0

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88

0.5

173

0.4

231

0.1

166

0.2

207

0

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68

0.3

109

0

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4

0.1

72

0

.30

2

0.2

152

0.1

728

0

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41

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119

0.1

697

0.2

983

0.5

217

0

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274

0

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55

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07

0.6

05

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3

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084

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64

0.5

017

0.5

995

0.3

698

0.2

313

0

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55

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552

0

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5

0.4

82

0

.24

8

0.2

263

0.4

856

0

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95

0.2

218

0.4

761

0.2

451

0.3

286

0

.43

9 0

.52

94

0.3

19

0.4

26

0

.51

5

0.3

207

0.4

301

0

.51

73

0.3

152

0.4

211

0.5

097

0.5

279

0

.24

25

0.4

307

0

.51

3

0.2

35

0

.41

7

0.5

142

0.2

366

0

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16

0.5

061

0.2

337

0.4

145

0.4

938

0

.56

26

0.3

787

0

.48

0.5

46

0

.36

8

0.4

816

0.5

491

0

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08

0.4

743

0.5

398

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64

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9

2.9

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4

4.0

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3.9

1

3.8

6

To

tal

per

centa

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of

erro

r 2

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2.2

9

3.9

3

Tab

le 6

.3(b

): C

om

par

ison o

f re

sult

s am

ong E

xper

imen

tal,

FE

A, T

heo

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cal

and S

PC

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for

esti

mat

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of

rela

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dep

th (

fix

ed-f

ixed

bea

m)



130


The present Chapter elaborates a noble damage detection procedure for structures under

traversing mass using statistical process control approach. The method has been divided

into two parts namely training and monitoring. The concept of control chart is introduced

to know the existence of damages in the structures. The control charts for the cantilever,

simply supported and fixed-fixed beam structures are represented in Figures 6.4(a), 6.4(b),

and 6.4(c) respectively. From the analysis of control chart, one can confirm whether the

structure is in damaged or healthy states.

The statistical process control method based on the perception of autoregressive (AR)

model is analyzed in time series domain for the fault detection in damaged structures. The

coefficients of AR model or damage sensitive features are determined using IBM SPSS

software package which provides the information about the vibration response data of the

structure. The magnitude in the actual variation in the AR model coefficients have been

computed statistically (Fisher Criterion) for both the damaged and healthy states of the

structures. The greater Fisher criterion values provide the information about the abrupt rise

in the response of the structures and thus able to recognize the relative crack locations.

After the damage localization process is over, and then the quantification of crack

severities is started. The stiffness vectors for each element of the damaged and undamaged

structures are determined using the fourth order statistical moment method. The central

differential scheme has been approached to calculate the strain values of the predicted

crack elements. The residuals vector have been obtained by considering the simulated

statistical moment with stiffness parameter and the actual statistical moment for the

measured displacement response of the structures. The information about the stiffness

parameters are obtained by minimizing the two statistical moments using the least square

algorithm. From the stiffness parameters, the damage severities are calculated. The results

obtained regarding the relative crack locations and crack depth from the statistical process

control approaches are represented in Tables 6.1 to 6.3 with FEA and theoretical methods

to verify the exactness of the proposed method. The comparison of results among various

methods for the determination of relative crack location are represented in Table 6.1(a) for

cantilever beam, Table 6.2(a) for simply supported beam and Table 6.3(a) for fixed-fixed

beam respectively, while those for relative crack depth the results are represented in

Tables 6.1(b), 6.2(b), 6.3(b) for cantilever, simply supported and fixed-fixed beam



131

respectively. It has been observed that the results obtained from SPC approach vary with

an average variation about 2.02 % and 3.93% with FEA and experiments respectively. So

the proposed method can be effectively applied to identify and quantify the cracks in the

structures.

132

Chapter 7

COMBINED HYBRID NEURO-

AUTOREGRESSIVE MODEL FOR

FAULT DETECTION IN BEAM

STRUCTURES SUBJECTED TO

MOVING MASS

7.1 Introduction

The use of mathematical model to explain the characteristics of physical phenomenon can

be well established. It is sometimes feasible to develop a method based on knowledge

based physical laws. The knowledge based may be some pattern recognition problems. In

fact no phenomenon is completely deterministic it’s because of the occurrence of

unknown factors. The problems based on statistical pattern recognition have recently

appeared as a promising tool to for automatic structural damage estimation. It is required

to implement some knowledge based theory in statistics based method and neural

networks approach for condition monitoring of structures. Numerous patterns recognition

algorithms based on statistical process control and neural network approaches are

developed for the automatic damage identification in structures. In this study, a combined

neural networks (Hybridisation of JRNNs and ERNNs) and autoregressive process

(Statistics process control method) based approach is developed for the structural damage

identification for beam type structures under moving mass. The efficiency and exactness

of the developed approach are also analyzed.

Chapter 7 Combined Hybrid Neuro-Autoregressive Model for Fault Detection

in Beam Structures Subjected to Moving Mass

133

7.2 Development of combined hybrid neuro-autoregressive

model for damage detection in beam type structures

subjected to moving mass

The integrated approach, hybrid neuro autoregressive model, is proposed for structural

damage detection based on the analyses of the autoregressive model and recurrent neural

networks. The autoregressive process is based on the domain of statistical process control

(SPC) method. Before the formulation of the combined hybrid-neuro autoregressive model

based approach, the RNNs (Chapter-5) and autoregressive (Chaptr-6) based methods are

developed. The objective of combining the two models (neural networks and

autoregressive models) is to refine the results obtained from the individual model and to

enhance the accuracy of the new developed method. The simple architecture of the

combined hybrid neuro-autoregressive model is represented in Figure 7.1.

Figure 7.1: Architecture of Hybrid neuro autoregressive model

AR Model

RCL-1

RCD-1

RCL-2

RCD-2

RCL-3

RCD-3

Displacement time histories

data of structures

NNs model

RCL-1

RCD-1

RCL-2

RCD-2

RCL-3

RCD-3

RD-1

RD-2

RD-3

RD-4

v (m/s)

M (kg)



134

In the proposed model, the autoregressive and NNs models are combined together to carry

out the fault detection procedures. Initially, the autoregressive analysis is carried out to

determine the relative crack locations and depth. The procedures to determine the relative

crack locations and depth using the autoregressive model are already explained in

Chapter-6. The same procedures are maintained here to find out the crack locations and

depth. Once the outputs are obtained from the autoregressive model, then these are

immediately fed to the neural network model as input parameters. The present neural

networks model has now some additional inputs apart from their original inputs.

The neural networks model is a recurrent neural network model. The architecture of the

present network model is same as that of the previous network model (Figure 5.8,

Chapter-5); the only difference is the additional inputs from the autoregressive model. The

present RNNs model, the hybridised model of JRNNs and ERNNs has 12 numbers of

input parameters to the network model. The other architectural issues and mechanism are

same as that in Figure 5.8 (Chapter-5).

The net input to the proposed network model is-

1

3 2 2, 3 3

2 1

St t t

j j j j j

j

w

(Chapter-5, equation- 5.23).

The netjt = 3

t

j = t

j =f (netjt) (Chapter-5, equation-5.24)

The netkt = 1

3 3,

3 1

St t

j j k k

j

w

(Chapter-5, equation-(5.25)

The net output of the proposed network is given by- ( )t t

k kg net . (Chapter-5, equation-

5.26)

Where, the symbols have their usual meanings as in Chapter-5.

The proposed neural network model has been also trained with Levenberg-Merquardt

algorithm with the same training procedures as in Chapter-5. The training processes have

been carried out using the sum square error function. The various relative crack positions

and depth are calculated.



135

Table 7.1(a): Comparison of results among FEA, Theoretical and hybrid approach of RNNs, and

SPC methods for prediction of relative crack locations (cantilever beam)

FEA Theoretical Hybrid RNNs and SPC


0.1625 0.3163 0.4318 0.1607 0.3124 0.4264 0.1600 0.3108 0.4241

0.2524 0.3928 0.4717 0.2500 0.3881 0.4664 0.2482 0.3855 0.4632

0.3672 0.5254 0.6436 0.3630 0.5198 0.6357 0.3619 0.5179 0.6335

0.4017 0.5015 0.6525 0.3967 0.4957 0.6447 0.3962 0.4938 0.6416

0.2225 0.4525 0.6727 0.2200 0.4474 0.6643 0.2189 0.4454 0.6622

0.4216 0.5272 0.8025 0.4176 0.5216 0.7938 0.4151 0.5182 0.7911

0.4415 0.582 0.743 0.4367 0.5759 0.7345 0.4354 0.5733 0.7307

0.517 0.584 0.646 0.5111 0.5776 0.6384 0.5088 0.5746 0.6344

0.6016 0.7215 0.8125 0.5962 0.7133 0.8027 0.5931 0.7092 0.7999

0.3726 0.5775 0.7755 0.3691 0.5709 0.7655 0.3666 0.5677 0.7619



Table 7.1(b): Comparison of results among FEA, Theoretical and hybrid approach of RNNs, and

SPC methods for prediction of relative crack depth (cantilever beam)



0.1425 0.2515 0.2728 0.1406 0.2488 0.2697 0.1398 0.2474 0.2681

0.2015 0.283 0.302 0.1991 0.2803 0.2988 0.1980 0.2786 0.2969

0.2065 0.305 0.3565 0.2041 0.3022 0.3522 0.2032 0.3007 0.3509

0.2455 0.2915 0.152 0.2431 0.2882 0.1501 0.2419 0.2875 0.1497

0.308 0.408 0.505 0.3047 0.4038 0.4989 0.3037 0.4022 0.4964

0.416 0.453 0.486 0.4112 0.4474 0.4814 0.4096 0.4465 0.4782

0.512 0.555 0.578 0.5059 0.5485 0.5712 0.5039 0.5463 0.5696

0.4565 0.528 0.544 0.4513 0.5217 0.5379 0.4496 0.5200 0.5354

0.3915 0.448 0.576 0.3867 0.4428 0.5703 0.3859 0.4414 0.5677

0.335 0.412 0.562 0.3307 0.4076 0.5559 0.3300 0.4063 0.5528





136

Table 7.2(a): Comparison of results among Experimental, FEA, Theoretical and hybrid approach

of RNNs, and SPC methods for prediction of relative crack locations (simply supported beam)

Experimental Theoretical FEA Hybrid RNNs and

SPC

rcl-1 rcl-2 rcl-3 rcl-1 rcl-2 rcl-3 rcl-1 rcl-2 rcl-3 rcl-1 rcl-2 rcl-3

0.2250 0.3750 0.4464 0.2185 0.3643 0.4350 0.2199 0.3669 0.4360 0.2166 0.3605 0.4315

0.2429 0.3857 0.4500 0.2362 0.3758 0.4376 0.2371 0.3778 0.4397 0.2336 0.3711 0.4344

0.3071 0.4929 0.5571 0.2984 0.4796 0.5410 0.2997 0.4818 0.5439 0.2971 0.4763 0.5384

0.3929 0.4857 0.5286 0.3806 0.4734 0.5141 0.3839 0.4742 0.5158 0.3789 0.4688 0.5112

0.4357 0.5286 0.6071 0.4242 0.5133 0.5911 0.4254 0.5163 0.5938 0.4194 0.5098 0.5879

0.5071 0.5857 0.6571 0.4926 0.5680 0.6401 0.4960 0.5715 0.6429 0.4876 0.5630 0.6345

0.5929 0.6643 0.7714 0.5768 0.6450 0.7508 0.5793 0.6512 0.7557 0.5736 0.6392 0.7439

0.6857 0.7571 0.8286 0.6666 0.7363 0.8059 0.6713 0.7416 0.8124 0.6622 0.7292 0.7976

0.3143 0.6286 0.8071 0.3053 0.6111 0.7841 0.3070 0.6164 0.7919 0.3036 0.6046 0.7779

0.2357 0.5214 0.8000 0.2290 0.5074 0.7790 0.2306 0.5098 0.7829 0.2276 0.5021 0.7706

Average percentage of

error

2.83 2.73 2.68 2.27 2.18 2.17 3.53 3.67 3.44

Total percentage of error 2.74 2.2 3.54

Table 7.2(b): Comparison of results among Experimental, FEA, Theoretical and hybrid approach

of RNNs, and SPC methods for prediction of relative crack depth (simply supported beam)

Experimental Theoretical FEA Hybrid RNNs and SPC

rcd-1 rcd-2 rcd-3 rcd-1 rcd-2 rcd-3 rcd-1 rcd-2 rcd-3 rcd-1 rcd-2 rcd-3

0.252 0.324 0.462 0.2444 0.3151 0.4492 0.2466 0.3169 0.4530 0.2433 0.3132 0.4449

0.334 0.576 0.482 0.3254 0.5608 0.4683 0.3271 0.5648 0.4712 0.3213 0.5550 0.4643

0.375 0.405 0.315 0.3651 0.3944 0.3058 0.3661 0.3970 0.3076 0.3613 0.3910 0.3044

0.156 0.526 0.294 0.1519 0.5127 0.2862 0.1522 0.5147 0.2869 0.1508 0.5085 0.2828

0.452 0.357 0.589 0.4396 0.3481 0.5729 0.4431 0.3493 0.5744 0.4359 0.3449 0.5664

0.551 0.426 0.217 0.5357 0.4150 0.2109 0.5404 0.4164 0.2119 0.5303 0.4114 0.2088

0.268 0.378 0.465 0.2608 0.3676 0.4520 0.2623 0.3690 0.4540 0.2582 0.3648 0.4479

0.295 0.472 0.496 0.2873 0.4594 0.4819 0.2883 0.4606 0.4851 0.2844 0.4553 0.4785

0.445 0.353 0.592 0.4335 0.3431 0.5750 0.4345 0.3447 0.5791 0.4293 0.3403 0.5703

0.512 0.202 0.401 0.4990 0.1965 0.3893 0.5009 0.1976 0.3920 0.4944 0.1950 0.3858

Average percentage of error 2.68 2.74 2.83 2.18 2.2 2.27 3.57 3.46 3.68

Total percentage of error 2.75 2.22 3.57



137

Table 7.3(a): Comparison of results among FEA, Theoretical and hybrid approach of RNNs, and

SPC methods for prediction of relative crack locations (fixed-fixed beam)



0.1486 0.2179 0.3257 0.1471 0.2158 0.3229 0.1461 0.2142 0.3205

0.1843 0.2543 0.3236 0.1822 0.2518 0.3197 0.1811 0.2508 0.3189

0.2179 0.2971 0.3771 0.2156 0.2945 0.3731 0.2146 0.2930 0.3710

0.2707 0.3229 0.4129 0.2675 0.3196 0.4085 0.2665 0.3186 0.4063

0.4300 0.5025 0.5775 0.4248 0.4969 0.5710 0.4225 0.4956 0.5690

0.4429 0.5500 0.6143 0.4383 0.5443 0.6083 0.4356 0.5429 0.6048

0.4571 0.5929 0.7143 0.4518 0.5864 0.7062 0.4493 0.5837 0.7034

0.5571 0.6357 0.8214 0.5502 0.6285 0.8129 0.5477 0.6257 0.8079

0.3357 0.5571 0.7714 0.3318 0.5477 0.7635 0.3304 0.5485 0.7605

0.3786 0.6571 0.7286 0.3725 0.6497 0.7203 0.3727 0.6474 0.7171



Table 7.3(b): Comparison of results among FEA, Theoretical and hybrid approach of RNNs, and

SPC methods for prediction of relative crack depth (fixed-fixed beam)



0.2520 0.3240 0.4620 0.2495 0.3211 0.4580 0.2479 0.3190 0.4541

0.3340 0.5760 0.4820 0.3305 0.5704 0.4775 0.3284 0.5664 0.4755

0.3750 0.4050 0.3150 0.3708 0.4010 0.3111 0.3685 0.3979 0.3103

0.1560 0.5260 0.2940 0.1541 0.5195 0.2905 0.1532 0.5173 0.2894

0.4520 0.3570 0.5890 0.4467 0.3525 0.5820 0.4445 0.3515 0.5809

0.5510 0.4260 0.2170 0.5443 0.4208 0.2145 0.5425 0.4197 0.2141

0.2680 0.3780 0.4650 0.2651 0.3737 0.4604 0.2636 0.3720 0.4584

0.2950 0.4720 0.4960 0.2923 0.4665 0.4910 0.2900 0.4644 0.4888

0.4450 0.3530 0.5920 0.4405 0.3487 0.5859 0.4384 0.3480 0.5829

0.5120 0.2020 0.4080 0.5071 0.1998 0.4036 0.5043 0.1993 0.4024





138


The implementation of damage detection scheme is often consigned to as structural health

monitoring process. This implementation process involves the characterization of potential

damage scenarios for the structural system. The extraction and analysis of damage features

determine the actual state of the system. Many damage assessment approaches have been

proposed. In this analysis a combined hybrid neuro-autoregressive model based approach

has been developed to locate the positions and quantify the severities of cracks in the

structure. The purpose of the combining the two methods is to improve the results

obtained from the individual methods.

The architectural scheme of the proposed integrated method is represented in Figure 7.1.

The autoregressive model is trained with the displacement time history data obtained from

the response of the structure. The output parameters from the autoregressive models are

fed to the RNNs model as input parameters along with the network’s normal input

parameters. The training procedure has been performed by employing the Levenberg-

Merquardt with 1000 number of iterations. The results obtained regarding the relative

crack depth from the present integrated approach are represented in Tables 7.1(a) for

cantilever,7.2(a) for simply supported and 7.3(a) for fixed-fixed beam structures

respectively, while those related to relative crack depth are represented in Tables

7.1(b),7.2(b) and 7.3(b) for cantilever, simply supported and fixed-fixed structures

respectively.

From analyse of results, it has been observed that the average variation of results between

the experimental and integrated approach of RNNs, and AR methods are near about 3.5%,

while those between the FEA and integrated approach of RNNs, and AR methods are near

about 1.5%. It has been concluded that the integrated approach of combined hybrid neuro-

autoregressive method predicts better results as comparison to RNNs and SPC

(autoregressive process) methods individually.

139

Chapter 8

EXPERIMENTAL ANALYSIS OF

DAMAGED STRUCTURES

SUBJECTED TO TRANSIT MASS

8.1 Introduction

Recently, structures under the action of moving objects are an emergent topic in the field

of structural dynamics. In most of the cases, the theoretical analysis or FEA does not

ensure the actual responses of structures precisely. To overcome the above limitations, real

time experimental techniques are necessary to study the actual responses of the structures.

The current Chapter addresses the detailed experimental procedures to determine the

responses of the beam structures subjected to moving mass at different end conditions.

8.2 Experimental Procedure

An experimental set-up has been developed in the laboratory for the moving mass-

structure systems. To verify the theoretical-numerical and FEA solutions, experiments

have been conducted for the different beam structures (cantilever, simply supported and

fixed-fixed beams) under traversing mass in the laboratory. The experimental set-up for

the transit mass-structure system are shown in Figures 8.1 (cantilever beam), 8.2 (simply

supported beam) and 8.3 (fixed-fixed beam). The different components of the

experimental set-up with machine specification are illustrated in Table 8.1. The

experimental procedures are carried out to determine the responses of the structures at

different position of the transit mass and also at the specified locations of the beam

specimen. The different components of the experimental set up are arranged as in the

Figures 8.1, 8.2 and 8.3. The mild steel beam specimens are considered for the

experimental analysis. The dimensions of the beam structures remain same as those of

theoretical-numerical analysis. The cracks are made at the appropriate positions of the

beam using wire EDM machine (Figure 8.4).

Chapter 8 Experimental Analysis of Damaged Structures Subjected to Transit Mass

140

Sen

sors

Mo

vin

g m

ass

Dis

pla

y u

nit

Var

iac

AC

mo

tor

Mic

ro c

ontr

oll

er

Bre

ad b

oar

d

Ro

pe

Fig

ure

8.1

: E

xper

imen

tal

set

up f

or

canti

lever

bea

m


141

The mass was allowed to slide on the cracked beam structure by connecting it one end of

the rope and another end of the rope was appended to the pulley. The pulley which in turn

was fixed to the shaft of the motor as in the Figures 8.1, 8.2 and 8.3. The main power

supply is A.C. So that power is supplied to the components of the experimental set-up

such as motor, variac, monitor, data acquisition unit and micro controller. The traversing

mass is allowed to slide across the beam structures without slipping. The length between

Figure 8.2: Experimental set up for simply supported beam

Display unit

Variac

AC motor

Rope

Moving mass

Sensors

Micro controller

Bread board

Figure 8.3: Experimental set up for fixed-fixed beam


142

the traversing mass and pulley was adjusted in a suitable manner such that there is no

slackness in the rope. The motor has been set with screws in a wooden tool in a proper

way that it will not be dislocated during the operation. The knob of the variac is adjusted

accurately to get the requisite constant speed of the traversing object and the best probable

precision in measurements. Intensive cares have been taken to place the variac knob to

obtain the required speed of the traversing mass. The speed of the traversing object is

assumed to be uniform while moving across the beam. Ultrasonic sensors are placed

below the beam structures at different positions through microcontroller and the

microcontroller is connected to the monitor (Display unit) through data acquisition unit.

The dynamic deflections of the beam structures are recorded through the sensors and

presented on the monitor. The average readings of dynamic deflections of the beam

structures at the different positions of the traversing mass are determined through the

sensors. Several numbers of tests are conducted to find out the deflections of the damaged

beam structures (cantilever, simply supported and fixed-fixed) with numerous speeds and

weight of the traversed mass.

Figure 8.5: Microcontroller (Audrino MEGA)

Figure 8.4: Ultrasonic sensor


143

Table 8.1: Specification of different components of experimental set up

S.N Items Specification

1 Power supply AC 220, 50Hz

2 Motor Type AC, 230V, 0.5 A,1/2 HP, RPM 1400, Type- M32.

Shaft diameter= 2.8 cm

3 Variac Input -230V, 50-60Hz,Output – 0-270V

4 Bread board Small Size Bread Board

5 Communication USB connection Serial Port

6 Microcontroller

Arduino MEGA , ATmega2560, 256 KB (ATmega2560)

Operating Voltage 5V, SRAM 8 KB (ATmega328)

Input Voltage (Recommended) 7-12V

Input Voltage (Limits) 6-20V, Digital Input Pins 54 (of Which 15 Provide

PWM Output), Analog Input Pins 14

7 Sensors Ultrasonic Range Finder Sensor Distance Measuring Range: 1cm to 400cm

8 Moving Mass 1 kg and 2 kg.

9 Specimen Mild steel

10 Cantilever beam Size=100cm×3.9cm×0.5cm, 1,2,3 =0.6, 0.25, 0.45 and 0.3, 0.55, 0.4

1,2,3 =0.25, 0.45, 0.65 and 0.5, 0.65, 0.85. v = 4.38 m/s and 5.73 m/s

11 Simply supported

beam

Size=140cm×4.9cm×0.5cm, 1,2,3 =0.2,0.3,0.4 and 0.35,0.45,0.55.

1,2,3 =0.2857, 0.5, 0.7143 and 0.1786, 0.3571, 0.5714. v = 4.38 m/s and

5.73 m/s

12 Fixed-fixed

beam

Size=140cm×4.9cm×0.5cm, 1,2,3 =0.2,0.35,0.45 and 0.3,0.5,0.55.

1,2,3 =0.1429, 0.3214, 0.5357 and 0.25, 0.4286, 0.7143. v = 5.12 m/s and

6.17 m/s

13 Vibration shaker Bruel and Kjaer, Frequency range-5 to 10 KHz, Maximum bare table

acceleration-700m/s2.

14 Delta Tron

Accelerometer Frequency range -1 to 10 KHz. Sensitivity-10mv/g to 500mv/g.

Figure 8.6: Damaged portion of beam


144

8.3 Discussions and summary

The experiments are carried out for the different beam structures under moving mass in

the laboratory. The responses of the structures at different damage scenarios are studied.

The dynamic deflection of the structures at different positions of the moving object and

specified locations are recorded through the sensors and Audrino micro controller. The

measured beam deflection from the experimental analysis are compared with those of

theoretical and FEA. In the experimental verifications, similar observations are obtained

regarding the responses of the structures as those in theoretical analysis and FEA. The

variations of results obtained from experimental analysis are with an average error of near

about 3% and 5% with FEA (Chapter-4) and theoretical analysis (Chapter-3) respectively.

So the proposed methods in the theoretical-numerical solutions and FEA converge well

with the experimentations. The relative crack positions and crack depth of the damaged

beam structures from the experimental analysis are also converged well with the different

methods like FEA, theoretical, RNNs, SPC, and integrated approach of the RNNs and

SPC.

Figure 8.8: Bread board

Figure 8.7: Variac

145

Chapter 9

RESULTS AND DISCUSSION

9.1 Introduction

The present Chapter elaborates the descriptions about the systematic procedures followed

in this thesis. The responses of the structures subjected to traversing mass are studied. The

numerical methods like Runge-Kutta and Newmark’s integration methods are applied to

study the dynamic behaviour of the structures under moving mass. The different damage

detection procedures such as JRNNs, ERNNs, combined JRNNs and ERNNs, AR process

and the combined hybrid process of JRNNs, ERNNs, and AR methods are discussed here.

The efficiencies and exactness of each method are explained.

9.2 Analysis and results of different adopted methods

The present study has been started with the review of different approaches to analyze the

response of different structures and ended with different methods to identify faults in

structures. The present dissertation is divided into nine Chapters. In each Chapter, the

different adopted methodologies are explained.

The motivation behind the research work, the objectives and novelty of the proposed

work, the layout of the entire thesis are explained in Chapter-1. The Chapter-2 (literature

review) explains about the motivate works carried out by different researcher, engineers

and scientist in the field of vibration and structural dynamics. The applications of various

numerical, FEA, experimental and soft computing methods have been also studied. From

the analysis of literature reviews, it gives us the knowledge gap between the previous and

present studies. Keeping in mind the knowledge gap between the past and present

analysis, the different techniques such as RNNs, SPC and the integrated approach of

RNNs, and SPC methods are applied for the crack detection in structures. The current

thesis is mainly organised into two categories i.e. determination of responses of structures

and detection of faults in structures.

In Chapter-3, the theoretical-numerical solutions of the multi-cracked structures subjected

to moving mass are analyzed. The definition and formulation of the proposed problem is

Chapter 9 Results and Discussion

146

also discussed here. The representation of multi-cracked cantilever beam (Figure 3.1),

simply supported beam (Figure 3.16), fixed-fixed beam (3.31) under traversing mass are

shown. The dynamic responses of the structures are investigated at different damage

configurations, moving mass and moving speeds of the structural systems. The solution of

the governing equation (3.18) has been solved with a numerical procedure of fourth order

Runge-Kutta method in MATLAB environment. The detailed response analyses of the

structures subjected to transit mass have been explained in Figures 3.2-3.11 (cantilever

beam), 3.17-3.26 (simply supported beam) and 3.32-3.41 (fixed-fixed beam). The 3-D

graphs for cantilever beam (Figures 3.12-3.15), simply supported beam (Figures 3.27-

3.30), fixed-fixed beam (Figures 3.43-3.45) are also explained at varying moving mass

and speed. The existence of cracks can be also known from the measured dynamic

deflections of the structures (Figures 3.46, 3.47 and 3.48). From the response analyses of

the structures, the consequence of parameters like crack locations, crack depth, moving

speed and moving mass are investigated on the response of the structures. The comparison

studies have been also carried out between undamaged and damaged structures under

traversing object.

In Chapter-4, the finite element analysis (FEA) using commercial ANSYS

WORKBENCH 2015 has been applied to determine the responses of the beam type’s

structures under transit mass. The transient dynamic analysis approach using the full

method is adopted in ANSYS WORKBENCH 2015. The computational approach

implemented in FEA is the Newmark’s integration method. The different steps involved in

the full method transient dynamic analysis are also explained. Before analyzing the

transient dynamic analysis, the modal analyses are performed to find out the natural

frequencies and mode shapes of the structures. The transit mass interaction dynamics of

the structural system is shown in Figure 4.5 (cantilever beam) and the magnified view of

the crack zone is shown in Figure 4.3. The frequencies ratios of the damaged structures are

represented in Tables 4.1, 4.4 and 4.7 for the cantilever, simply supported and fixed-fixed

beam structures. The different modal behaviour of the structures is also represented in

Figure 4.4(cantilever beam), Figure 4.7(simply supported beam) and Figure 4.9 (fixed-

fixed beam). Similar observations are made regarding the response of the structures in

FEA as those in numerical analysis.

In Chapter-8, the experimental models have been developed for each of the structure

subjected to traversing object in the laboratory. The laboratory set-ups for the structural


147

systems under moving mass are shown in Figures 8.1, 8.2 and 8.3 for cantilever, simply

supported and fixed-fixed structures respectively. The different equipments of the

experimental set-ups, their specifications and functions are described in Table 8.1. The

experimental procedures have been already explained in Chapter-8. The responses of the

structures are also determined by laboratory tests. The purpose of the laboratory tests is to

verify the accuracy and exactness of FEA and numerical analysis.

The detailed analyses regarding the response of the structures have been illustrated in

Chapters 4, 5 and 8. The accuracy of each method has been compared with each other.

The comparison of results among experiments, FEA and numerical analysis for the

response of structures are represented in graphical way in Figure 4.6 (damaged cantilever

beam), Figure 4.8 (damaged simply supported beam) and Figure 4.10 (damaged fixed-

fixed beam). The variation of results between the experimental and numerical analyses are

expressed in Tables 3.1,3.2 and 3.3 for cantilever, simply supported and fixed-fixed beam

structures under moving mass respectively. Similarly, the disparity of results among the

laboratory tests, FEA and numerical analysis are demonstrated in Tables 4.2-4.3

(cantilever beam), Tables 4.5-4.6 (simply supported beam) and Tables 4.8-4.9 (fixed-fixed

beam. From analyses of results obtained from the different methods, it has been observed

that there are variations of results near about 5% between the experimental and numerical

analyses, while those between the experimental and FEA are near about 2.9%. So the

applied numerical and FEA methods converge well with the experimental works.

The novel damage detection procedures have been also developed for finding out the

faults in the structures. The fault identification methods are in the domain of neural

networks and statistical process control environments.

In Chapter-5, the rule- based recurrent neural networks (RNNs) methods are developed to

identify the faults in the damaged structural systems. The knowledge based Jordan’s

recurrent neural networks (JRNNs), Elman’s recurrent neural network (ERNNs) and the

combined approach of JRNNs, and ERNNs are mainly focused in this Chapter. The

Levenberg-Merquardt’s back propagation algorithm has been implemented to train the

proposed RNNs model with sum squared error. The different steps involved for organising

the training procedures of the proposed network using the Levenberg-Merquardt’s

algorithm are also explained. The architecture of the modified JRNNs, ERNNs and the

combined approach of JRNNs, and ERNNs are represented in Figures 5.6, 5.7 and 5.8

respectively. The detailed training procedures of the RNNs are already explained in


148

Chapter-5. 600 numbers of training patterns are generated for each of the damaged

structure and the networks are trained with 1000 numbers of iterations. The relative crack

locations and depth of each multi-cracked structure are predicted using the RNNs

techniques. The comparisons of results among experimental, JRNNs, ERNNs and

combined approach of JRNNs, and ERNNs for relative crack locations are represented in

Tables 5.2(a), 5.3(a) and 5.4(a) for cantilever, simply supported and fixed-fixed structures

respectively, while those for relative crack depth are expressed in Tables 5.2(b), 5.3(b) and

5.4(b) respectively. It has been remarked that the disparities of results between

experiments and JRNNs are an average near about 6.5%, while those with ERNNs and the

combined approach of JRNNs, and ERNNs are about 5.5% and 4.5% respectively.

In Chapter-6, the innovative damage identification process has been evolved in the domain

of statistical process control method using the time series analysis concepts. The theory of

control chart analysis is established to know the existence of cracks in the structures. The

concepts of autoregressive (AR) process are applied to identify the faults in structures. In

AR process, the training patterns are generated by considering both the damaged and

undamaged states of the structures. The complete training and monitoring process of the

AR model are previously explicated in Chapter-6. The coefficients of AR model represent

the damage sensitive features of the structures. The displacement-time data are considered

as the input parameters to the AR model. Using the hypothesis of Fisher’s criterion, the

locations of the damages have been identified. The fourth order statistical moment method

for stiffness vector and probability density function are established to find out the relative

crack depth. The comparisons of results among FEA, theoretical and SPCM methods for

relative crack locations are represented in Tables 6.1(a), 6.2(a) and 6.3(a) for cantilever,

simply supported and fixed-fixed beams respectively, while those for relative crack depth

are expressed in Tables 6.1(b), 6.2(b) and 6.3(b) respectively. It has been observed that the

variation of results between FEA and SPCM are near about 1.95%, while those between

experiments and SPCM are of 3.9% approximately.

In order to improve the results obtained from Chapters 5 and 6, a combined hybrid neuro-

autoregressive method has been elaborated in Chapter-7. The outputs from the AR model

are fed to hybridised JRNNs and ERNNs model as input to the newly developed network

model. The newly developed RNNs model has some additional inputs (outputs from AR

model) along with their usual inputs. The training and monitoring procedures of newly

developed model are already explained in Chapters-5 and 6. The comparison of results


149

between combined hybrid neuro-autoregressive process and experiments are average

errors of 3.5%, while those with FEA are of 1.55% approximately. The comparison of

different damage detection methods are shown in Figure 9.1. It has been concluded that

the hybrid-neuro AR process produce better results.

9.3 Summary

Several intelligent methods such as JRNNs, ERNNs, hybridised approach of JRNNs and

ERNNs, SPC method, and the integrated approach of combined hybrid neuro-

autoregressive methods have been implemented for fault detection in multi cracked

structures under moving mass. The accuracy and exactness of each intelligent method

have been compared with experiments, FEA and theory. It has been concluded that the

combined hybrid neuro-autoregressive method yields better results as comparison to other

results.

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10

Perc

en

ata

ge v

alu

e

Number of observations

JRNNs

ERNNs

Combined JRNNs and

ERNNsAR process

Hybrid neuro-AR process

Theory

FEA

Figure 9.1 Comparison of results among different damage

detection methods

150

Chapter 10

CONCLUSIONS AND SUGGESTION

FOR FURTHER RESEARCH

10.1 Introduction

In the present analysis, the vibration analyses of damaged structures subjected to transit

mass are studied. The responses of the multi-cracked structures with different boundary

conditions have been analyzed with different methods. The damage identification

procedures have been carried out using different soft computing approaches.

10.2 Contributions

The dynamic responses of beam type’s structures with multiple cracks subjected to

moving object are studied. The theoretical-numerical solutions of the beam structures

under transit mass has been formulated and consequently solved by applying Runge-Kutta

fourth order integration scheme. The FEA and experimental analysis have been carried out

to verify the applied numerical method. The FEA method (The full method transient

dynamic analysis) in ANSYS WORKBENCH 2015 domain has been implemented to find

out the responses of the structures subjected to transit mass. The laboratory tests for each

of the structural systems have been performed to check the accuracy of the proposed

computational and FEA methods. The responses of the structures subjected to transit mass

are determined with various damage configurations, moving speed and traversing mass of

the structural systems. The influences of different parameters like traversing mass, moving

speed, crack locations and depth on the response of the structures are also investigated.

The various intelligent methods such as RNNs, SPC, and integrated approach of RNNs

and SPC based methods have been applied as inverse problems for fault detection in the

damaged structures. Some knowledge based RNNs like JRNNs, ERNNs, and hybrid

approach of JRNNs and ERNNs are implemented to estimate the relative crack locations

and depth. The Levenberg-Merquardt’s algorithm has been implemented to train the

networks. The autoregressive (AR) process in SPC domain are applied for the prediction

Chapter 10 Conclusions and Suggestion for Further Research

151

of relative crack locations. The fourth order statistical moment with central differential

scheme is applied to quantify the crack severities. For the improvement of the above

methods and results, the RNNs and SPC methods are integrated to form a combine hybrid

neuro-autoregressive method. This hybrid neuro-autoregressive method has been used for

the prediction of relative crack locations and depth in the structures.

10.3 Conclusions

The dynamic analysis and fault detection of cracked structures under moving mass has

been studied. The influences of different parameters such as traversing mass, moving

speed, crack locations and depth on the dynamic responses of the structures under moving

mass have been investigated. The FEA and experimental verifications have been carried

out to check the accuracy and exactness of the proposed computational method (Runge-

Kutta method). It has been observed that the results obtained from the numerical analysis

are varying with error about 5% with experimental procedures and about 2% with FEA.

So the implemented computational method agrees well with FEA and experiments.

The relative crack locations and depth have been predicted using some knowledge based

intelligent methods like RNNs, statistical process controls (SPC), and the integrated

approach of the RNNs and SPC. The errors of result between experiments and JRNNs is

near about 6.5%, while those with ERNNs and the hybrid approach of JRNNs and ERNNs

are about of 5.5% and 4.4% respectively. The damage detection procedures have been

carried out using the concept of time series analysis in SPC domain. The autoregressive

(AR) process in SPC domain are applied for the prediction of relative crack locations. The

fourth order statistical moment with central differential scheme is applied to quantify the

crack severities. The disparities of results between the AR process and experiments are

with an average error of 3.95%, while that with FEA is 1.95% approximately. The results

obtained from the hybrid neuro-autoregressive method are converged with an error of

3.5% with experiments and 1.55% with FEA respectively.

It has been come to an end that the combined hybrid neuro-autoregressive gives the best

result as compared to other methods addressed in the current analogy. Using the neural

network and SPC controller, the online fault detection can be carried out for beam

structures under traversing mass.

Chapter 10 Conclusions and Suggestion for Further Research

152

10.4 Recommendation for future study

The problem may be extended in the following ways:

The theoretical-numerical solution of damaged structure subjected to moving mass

can be formulated under different foundation of structures.

The vibration analysis of damaged shaft under traversing mass is to be carried out.

The response analysis of structure subjected to transit mass at variable speed can

be found out.

The damage detection procedures can be developed using the integrated approach

of RNNs and Genetic algorithm, Statistical process control methods such as

autoregressive moving approach, Statistical moment based approach and some

natures inspired algorithm like multi Swarm Fruit Fly, Collaborative-Climb

Monkey and Climbing Hill algorithms.

153

Appendix

Appendix-A

Crack analysis on the vibration characteristics of cantilever beam:

A cracked cantilever beam (Fig A-1.) of length ‘L’, width ‘B’, thickness ‘H’ with multiple

cracks of crack depth ‘ 1,2,3d ’, at distance of ‘ 1,2,3L ’ from the fixed end is considered for the

analysis. The analysis of the cracked beam is carried out as follows

( , )u x t = Longitudinal vibration displacement functions of the crack section

( , )y x t = Transverse vibration displacement functions of the crack section.

One can define the normalized function for the cracked structure with multiple cracks in

normalized form as

1 1 2( ) cos( ) sin( )u uu x A K x A K x (A-1)




1 9 10 11 12( ) cosh( ) sinh( ) cos( ) sin( )y y y yy x A K x A K x A K x A K x (A-5)




The normalized form may be expressed as

1,2,3 1,2,3 /L L =Relative location of cracks.

/ , / , / , /x x L u u L y y L t t L

11 122 2

/ , , , , u u y yy

LE EIK L C C K C m AC m

The different values of ‘A’ ( ( 1,24))iA i can be determined from the different end

conditions.

The different end conditions of the cantilever beam are as follows

1 1 1(0) 0, (0) 0, (0) 0u y y (A-9)

Appendix

154

At the free end , then 1xx L xL

4 4 4(1) 0, (1) 0, (1) 0u y y (A-10)

At the crack segment

1 1 2 1 2 2 3 2 3 3 4 3( ) ( ), ( ) ( ), ( ) ( )u u u u u u (A-11)

1 1 2 1 2 2 3 2 3 3 4 3( ) ( ), ( ) ( ), ( ) ( )y y y y y y (A-12)

1 1 2 1 2 2 3 2 3 3 4 3( ) ( ), ( ) ( ), ( ) ( )y y y y y y (A-13)

1 1 2 1 2 2 3 2 3 3 4 3( ) ( ), ( ) ( ), ( ) ( )y y y y y y (A-14)

At the first crack segment also

1 11 1 2 1 1 111 2 1 1 1 12

( ) ( ) ( )( ) ( )

du L dy L dy LAE K u L u L K

dx dx dx

(A-15)

Similarly for the second and third crack sections, one can express

2 2 3 22 2 2 211 3 2 2 2 12

( )( ) ( )( ) ( )

dy Ldu L dy LAE K u L u L K

dx dx dx

(A-16)

3 33 3 4 3 3 311 4 3 3 3 12

( ) ( ) ( )( ) ( )

du L dy L dy LAE K u L u L K

dx dx dx

(A-17)

Equation (A-15) is for discontinuity due to axial deformation before and after the first

crack.

Similarly due to the discontinuity of slope to the before and after of the first crack section,

one can express

21 11 1 2 1 1 1

21 2 1 1 1 222

( ) ( ) ( )[ ( ) ( )]

d y L dy L dy LEI K u L u L K

dx dx dx

(A-18)

Similarly for the second and third crack sections, one can express

22 2 3 22 2 2 2

21 3 2 2 2 222

( )( ) ( )[ ( ) ( )]

dy Ld y L dy LEI K u L u L K

dx dx dx

(A-19)

23 33 3 4 3 3 3

21 4 3 3 3 222

( ) ( ) ( )[ ( ) ( )]

d y L dy L dy LEI K u L u L K

dx dx dx

(A-20)

By inversing the compliance matrix, the local flexibility matrix element ‘ ijK ’ may be

determined.

One can determine the compliance matrix element by considering the strain energy at the

crack location- 12

0

( )

d

ij

i j

C J a ddP P

(A-21)

Appendix

155

Where ( )J a = Strain energy density function

1iP =Axial force, 2iP =Bending moment

156

Bibliography

1. N. Sridharan and A. K. Mallik, “Numerical analysis of vibration of beams

subjected to moving loads”, Journal of Sound and Vibration, vol. 65, no. l, pp.

147-150, 1979.

2. S. A. Q. Siddiqui, M. F. Golnaraghi and G. R. Heppler, “Dynamics of a flexible

cantilever beam carrying a moving mass”, Nonlinear Dynamics, vol. 15, pp. 137-

154, 1998.

3. J. E. Akin and M. Mofld, “Numerical solution for response of beams with

moving mass”, Journal of Structural Engineering, vol. 115, pp. 120-131, 1989.

4. M. M. Stanisic and J. C. Hardin, “On the response of beams to an arbitrarily

number of concentrated moving masses”, Journal of the Franklin Institute,

vol. 287, no. 2, pp. 115-124, 1969.

5. M. Olsson, “On the fundamental moving load problem”, Journal of Sound and

Vibration, vol. 145, no. 2, pp. 299-307, 1991.

6. M. Moffid and J. E. Akin, “Discrete element response of beams with travelling

mass”, Advances in Engineering Structure, vol. 25, pp. 321-331, 1996.

7. H. C. Kwon, M. C. Kim and I. W. Lee, “Vibration control of bridges under

moving loads”, Computers & Structures, vol. 66, no. 4, pp. 473-480, 1998.

8. M. A. Mahmoud and M. A. Abouzaid, “Dynamic response of a beam with a

crack subject to a moving mass”, Journal of Sound and Vibration, vol. 256, no. 4,

pp. 591-603, 2002.

9. J. Li, M. Su and L. Fan, “Natural frequency of railway girder bridges under

vehicle loads”, Journal of Bridge Engineering, vol. 8, no. 4, pp. 199-203, 2003.

10. C. Bilello, L. A. Bergman and D. Kuchma, “Experimental investigation of a

small-scale bridge model under a moving mass”, Journal of Structural

Engineering, vol. 130, no. 5, pp. 799-804, 2004.

11. C. Bilello, L. A. Bergman, “Vibration of damaged beams under a moving mass:

theory and experimental validation”, Journal of Sound and Vibration, vol. 274,

pp. 567-582, 2004.

12. Y. B. Yang, C. W. Lin and J. D. Yau, “Extracting bridge frequencies from the

dynamic response of a passing vehicle”, Journal of Sound and Vibration, vol.

272, pp. 471-493, 2004.

13. M. Majka and M. Hartnett, “Effects of speed, load and damping on the dynamic

response of railway bridges and vehicles”, Computers and Structures, vol. 86, pp.

556-572, 2008.

14. A. Garinei and G. Risitano, “Vibrations of railway bridges for high speed trains

under moving loads varying in time”, Engineering Structures, vol. 30, pp. 724-

732, 2008.

Reference

157

15. Y. B. Yang and K. C. Chang, “Extracting the bridge frequencies indirectly from a

passing vehicle: Parametric study”, Engineering Structures, vol. 31, pp. 2448-

2459, 2009.

16. M. Dehestani, M. Mofid and A. Vafai, “Investigation of critical influential speed

for moving mass problems on beams”, Applied Mathematical Modelling, vol. 33,

pp. 3885-3895, 2009.

17. K. Liu, G. D. Roeck and G. Lombaert, “The effect of dynamic train–bridge

interaction on the bridge response during a train passage”, Journal of Sound and

Vibration, vol. 325, pp. 240-251, 2009.

18. D. M. Siringoringo and Y. Fujino, “Estimating bridge fundamental frequency

from vibration response of instrumented passing vehicle: Analytical and

experimental study”, Advances in Structural Engineering, vol. 15, no. 3, pp. 443-

460, 2012.

19. H. Xia, N. Zhang and W. W. Guo, “Analysis of resonance mechanism and

conditions of train-bridge system”, Journal of Sound and Vibration, vol. 297, pp.

810-822, 2006.

20. M. Majkaa and M. Hartnett, “Dynamic response of bridges to moving trains: A

study on effects of random track irregularities and bridge skewness”, Computers

and Structures, vol. 87, pp. 1233-1252, 2009.

21. S. -H. Ju, H. -T. Lin and J. -Y. Huang, “Dominant frequencies of train-induced

vibrations”, Journal of Sound and Vibration, vol. 319, pp. 247-259, 2009.

22. H. -I. Yoon, I. -S. Son and S. -J. Ahn, “Free vibration analysis of Euler-Bernoulli

beam with double cracks”, Journal of Mechanical Science and Technology, vol.

21, pp. 476-485, 2007.

23. M. A. Mahmoud, “Stress intensity factors for single and double edge cracks in a

simple beam subject to a moving load”, International Journal of Fracture, vol.

111, pp. 151-161, 2001.

24. G. Michaltsos, D. Sophianopoulos and A. N. Kounadis, “The effect of a moving

mass and other parameters on the dynamic response of a simply supported

beam”, Journal of Sound and Vibration, vol. 191, no. 3, pp. 357-362, 1996.

25. M. A. Mahmoud, “Effect of cracks on the dynamic response of a simple beam

subject to a moving load”, Journal of Rail and Rapid Transit, vol. 215, pp. 206-

215, 2001.

26. H. -P. Lin and S. -C. Chang, “Forced responses of cracked cantilever beams

subjected to a concentrated moving load”, International Journal of Mechanical

Sciences, vol. 48, pp. 1456-1463, 2006.

27. Ouyang, H., “Moving-load dynamic problems: A tutorial (with a brief

overview)”, Mechanical Systems and Signal Processing, 2011, 25, 2039-2060.

Reference

158

28. Ariaei, A., Ziaei-Rad, S. and Ghayour, M, “Vibration analysis of beams with

open and breathing cracks subjected to moving masses”, Journal of Sound and

Vibration, 2009, 326, 709-724.

29. S. E. Azam, M. Mofid and R. A. Khoraskani, “Dynamic response of Timoshenko

beam under moving mass”, Scientia Iranica A, vol. 20, no. 1, pp. 50-56, 2013.

30. G. T. Michaltsos and A. N. Kounadis, “The effects of centripetal and Coriolis

forces on the dynamic response of light bridges under moving loads”, Journal of

Vibration and Control, vol. 7, pp. 315-326, 2001.

31. Y. Pala and M. Reis, “Dynamic response of a cracked beam under a moving mass

load”, Journal of Engineering Mechanics, vol. 129, pp. 1229-1238, 2013.

32. M. Reis and Y. Pala, “Vibration of a cracked cantilever beam under moving mass

load”, Journal of Civil Engineering and Management, vol. 18, no. 1, pp. 106-113,

2012.

33. X. Shi, C. S. Cai and S. Chen, “Vehicle induced dynamic behavior of short-span

slab bridges considering effect of approach slab condition”, Journal of Bridge

Engineering, vol. 13, pp. 83-92, 2008.

34. E. Esmailzadeh and M. Ghorashi, “Vibration analysis of a Timoshenko beam

subjected to a travelling mass”, Journal of Sound and Vibration, vol. 199, no. 4,

pp. 615-628, 1997.

35. H. P. Lee, “The dynamic response of a Timoshenko beam subjected to a moving

mass”, Journal of Sound and Vibration, vol. 198, no. 2, pp. 249-256, 1996.

36. E. Khalily, M. E. Golnaraghi and G. R. Heppler, “On the dynamic behaviour of a

flexible beam carrying a moving mass”, Nonlinear Dynamics, vol. 5, pp. 493-

513, 1994.

37. M. Mofid and M. Shadnam, “On the response of beams with internal hinges,

under moving mass”, Advances in Engineering Software, vol. 31, pp. 323-328,

2000.

38. A. O. Cifuentes, “Dynamic response of a beam excited by a moving mass”, Finite

Elements in Analysis and Design, vol. 5, pp. 237-24, 1989.

39. A. Nikkhoo, F. R. Rofooei and M. R. Shadnam, “Dynamic behavior and modal

control of beams under moving mass”, Journal of Sound and Vibration, vol. 306,

pp. 712-724, 2007.

40. D. A. Grant, “The effect of rotary inertia and shear deformation on the frequency

and normal mode equations of uniform beams carrying a concentrated mass”,

Journal of Sound and Vibration, vol. 57, no. 3, pp. 357-365, 1978.

41. D. Thambiratnam and Y. Zhuge, “Dynamic analysis of beams on an elastic

foundation subjected to moving loads”, Journal of Sound and Vibration, vol. 198,

no. 2, pp. 149-169, 1996.

42. G. V. Rao, “Linear dynamics of an elastic beam under moving loads”, Journal of

Vibration and Acoustics, vol. 122, pp. 281-289, 2000.

Reference

159

43. A. Yavari, M. Nouri and M. Mofid, “Discrete element analysis of dynamic

response of Timoshenko beam under moving mass”, Advances in Engineering,

vol. 33, pp. 143-153, 2002.

44. M. Abu-Hilal, “Dynamic response of a double Euler–Bernoulli beam due to a

moving constant load”, Journal of Sound and Vibration, vol. 297, pp. 477-491,

2006.

45. H. P. Wang and J. L. K. Zhang, “Vibration analysis of the maglev guideway with

the moving load”, Journal of Sound and Vibration, vol. 305, pp. 621-640, 2007.

46. J. Yang, Y. Chen, Y. Xiang and X. L. Jia, “Free and forced vibration of cracked

inhomogeneous beams under an axial force and a moving load”, Journal of Sound

and Vibration, vol. 312, pp. 166-181, 2008.

47. T. Yan, S. Kitipornchai, J. Yang and X. Q. He, “Dynamic behaviour of edge-

cracked shear deformable functionally graded beams on an elastic foundation

under a moving load”, Composite Structures, vol. 93, pp. 2992-3001, 2011.

48. M. Shafiei and N. Khaji, “Analytical solutions for free and forced vibrations of a

multiple cracked Timoshenko beam subject to a concentrated moving load”, Acta

Mechanica, vol. 221, pp. 79-97, 2011.

49. S. -I. Suzuki, “Dynamic behaviour of a finite beam subjected to travelling

loads with acceleration”, Journal of Sound and Vibration, vol. 55, no. l, pp.

65-70, 1977.

50. T. R. Hamada, “Dynamic analysis of a beam under a moving force: a

double Laplace transform solution”, Journal of Sound and Vibration, vol. 74,

no. 2, pp. 221-233, 1981.

51. H. P. Lee and T. Y. Ng, “Dynamic response of a cracked beam subject to a

moving load”, Acta Mechanica, vol. 106, pp. 221-230, 1994.

52. M. Ichikawa, Y. Miyakawa and A. Matsuda, “Vibration analysis of the

continuous beam subjected to a moving mass”, Journal of Sound and Vibration,

vol. 230, no. 3, pp. 493-506, 2009.

53. J. J. Wu, A. R. Whittaker and M. P. Cartmell, “Dynamic responses of structures

to moving bodies using combined finite element and analytical methods”,

International Journal of Mechanical Sciences, vol. 43, pp. 2555-2579, 2001.

54. A. K. Mallik, S. Chandra and A. B. Singh, “Steady-state response of an

elastically supported infinite beam to a moving load”, Journal of Sound and

Vibration, vol. 291, pp. 1148-1169, 2006.

55. K. Aydin, “Vibratory characteristics of Euler-Bernoulli beams with an arbitrary

number of cracks subjected to axial load”, Journal of Vibration and Control, vol.

14, no. 4, pp. 485-510, 2008.

56. R. Sieniawska, P. Sniady and S. Zukowski, “Identification of the structure

parameters applying a moving load”, Journal of Sound and Vibration, vol. 319,

pp. 355-365, 2009.

Reference

160

57. T. Yan and J. Yang, “Forced vibration of edge-cracked functionally graded

beams due to a transverse moving load”, Procedia Engineering, vol. 14, pp.

3293-3300, 2011.

58. G. Chen, L. Qian and Q. Yin, “Dynamic analysis of a Timoshenko beam

subjected to an accelerating mass using spectral element method”, Shock and

Vibration, 2014, Article ID 768209.

59. R. Zarfam and A. R. Khaloo, “Vibration control of beams on elastic foundation

under a moving vehicle and random lateral excitations”, Journal of Sound and

Vibration, vol. 331, pp. 1217-1232, 2012.

60. C. Johansson, C. Pacoste and R. Karoumi, “Closed-form solution for the mode

superposition analysis of the vibration in multi-span beam bridges caused by

concentrated moving loads”, Computers and Structures, vol. 119, pp. 85-94,

2013.

61. P. Lou and F. T. K. Au, “Finite element formulae for internal forces of Bernoulli-

Euler beams under moving vehicles”, Journal of Sound and Vibration, vol. 332,

pp. 1533-1552, 2013.

62. P. Museros, E. Moliner and M. D. Martinez-Rodrigo, “Free vibrations of simply-

supported beam bridges under moving loads: Maximum resonance, cancellation

and resonant vertical acceleration”, Journal of Sound and Vibration, vol. 332, pp.

326-345, 2013.

63. R. Zarfam, A. R. Khaloo and A. Nikkhoo, “On the response spectrum of

Euler–Bernoulli beams with a moving mass and horizontal support

excitation”, Mechanics Research Communications, vol. 47, pp. 77-83, 2013.

64. H. Azimi, K. Galal and O. A. Pekau, “A numerical element for vehicle–bridge

interaction analysis of vehicles experiencing sudden deceleration”, Engineering

Structures, vol. 49, pp. 792-805, 2013.

65. A. Cicirello and A. Palmeri, “Static analysis of Euler–Bernoulli beams with

multiple unilateral cracks under combined axial and transverse loads”,

International Journal of Solids and Structures, vol. 51, pp. 1020-1029, 2014.

66. H. Zhong, M. Yang and Z. Gao, “Dynamic responses of prestressed bridge and

vehicle through bridge-vehicle interaction analysis”, Engineering Structures, vol.

87, pp. 116-125, 2015.

67. P. A. Costa, A. Colaco, R. Calcada and A. S. Cardoso, “Critical speed of railway

tracks. Detailed and simplified approaches”, Transportation Geotechnics, vol. 2,

pp. 30-46, 2015.

68. C. Fu, “The effect of switching cracks on the vibration of a continuous beam

bridge subjected to moving vehicles”, Journal of Sound and Vibration, vol. 339,

pp. 157-175, 2015.

Reference

161

69. C. Fu, “Dynamic behavior of a simply supported bridge with a switching crack

subjected to seismic excitations and moving trains”, Engineering Structures, vol.

110, pp. 59-69, 2016.

70. H. Aied, A. González and D. Cantero, “Identification of sudden stiffness changes

in the acceleration response of a bridge to moving loads using ensemble empirical

mode decomposition”, Mechanical Systems and Signal Processing, vol. 66-67,

pp. 314-338, 2016.

71. J. Hino, T. Yoshimura, K. Konishi and N. Ananthanarayana, “A finite element

method vibration of a bridge prediction of the subjected to a moving vehicle

load”, Journal of Sound and Vibration, vol. 96, no. l, pp. 45-53, 1984.

72. G. R. Bhashyam and G. Prathap, “Galerkin finite element method for non-

linear beam vibrations”, Journal of Sound and Vibration, vol. 72, no. 2, pp.

191-203, 1980.

73. M. Olsson, “Finite element, modal co-ordinate analysis of structures subjected to

moving loads”, Journal of Sound and Vibration, vol. 99, no. l, pp. l-12, 1985.

74. T. Yoshimura, J. Hino, T. Kamata and N. Ananthanarayana, “Random vibration

of a non-linear beam subjected to a moving load: A finite element method

analysis,” Journal of Sound and Vibration, vol. 122, no. 2, pp. 317-329, 1988.

75. Y. H. Lin and M. W. Trethewey, “Finite element analysis of elastic beams

subjected to moving dynamic loads”, Journal of Sound and Vibration, vol. 136,

no. 2, pp. 323-342, 1990.

76. 76. T. P. Chang and Y. -N. Liu, “Dynamic finite element analysis of a

nonlinear beam subjected to a moving load”, International Journal of Solids

Structures, vol. 33, no. 12, pp. 1673-1688, 1996.

77. J. R. R1eker and M. W. Trethewey, “Finite element analysis of an elastic beam

structure subjected to a moving distributed mass train”, Mechanical Systems and

Signal Processing, vol. 13, no. 1, pp. 31-51, 1999.

78. M. -K. Songa, H. -C. Noh and C. -K. Choi, “A new three-dimensional finite

element analysis model of high-speed train-bridge interactions”, Engineering


79. S. S. Law and X. Q. Zhu, “Dynamic behavior of damaged concrete bridge

structures under moving vehicular loads”, Engineering Structures, vol. 26, pp.

1279-1293, 2004.

80. Y. B. Yang and C. W. Lin, “Vehicle–bridge interaction dynamics and potential

applications”, Journal of Sound and Vibration, vol. 284, pp. 205-226, 2005.

81. L. Kwasniewski, H. Li, J. Wekezer and J. Malachowsk, “Finite element analysis

of vehicle-bridge interaction”, Finite Elements in Analysis and Design, vol. 42,

pp. 950-959, 2006.

Reference

162

82. S. -H. Ju and H. -T. Lin, “A finite element model of vehicle–bridge interaction

considering braking and acceleration”, Journal of Sound and Vibration, vol. 303,

pp. 46-57, 2007.

83. C. I. Bajer and B. Dyniewicz, “Virtual functions of the space–time finite element

method in moving mass problems”, Computers and Structures, vol. 87, pp. 444-

455, 2009.

84. V. Kahya, “Dynamic analysis of laminated composite beams under moving loads

using finite element method”, Nuclear Engineering and Design, vol. 243, pp. 41-

48, 2012.

85. B. Dyniewicz, “Space-time finite element approach to general description of a

moving inertial load”, Finite Elements in Analysis and Design, vol. 62, pp. 8-17,

2012.

86. I. Esen, “A new finite element for transverse vibration of rectangular thin plates

under a moving mass”, Finite Elements in Analysis and Design, vol. 66, pp. 26-

35, 2013.

87. W. Zhang, C. S. Cai and F. Pan, “Finite element modeling of bridges with

equivalent orthotropic material method for multi-scale dynamic loads”,

Engineering Structures, vol. 54, pp. 82-93, 2013.

88. J. V. Amiri, A. Nikkhoo, M. R. Davoodi and M. E. Hassanabadi, “Vibration

analysis of a Mindlin elastic plate under a moving mass excitation by

eigenfunction expansion method”, Thin-Walled Structures, vol. 62, pp. 53-64,

2013.

89. S. H. Ju, “Improvement of bridge structures to increase the safety of moving

trains during earthquakes”, Engineering Structures, vol. 56, pp. 501-508, 2013.

90. F. B. Nejad, A. Khorram and M. Rezaeian, “Analytical estimation of natural

frequencies and mode shapes of a beam having two cracks”, International

Journal of Mechanical Sciences, vol. 78, pp. 193-202, 2014.

91. H. Aied and A. Gonzalez, “Theoretical response of a simply supported beam with

a strain rate dependant modulus to a moving load”, Engineering Structures, vol.

77, pp. 95-108, 2014.

92. J. Wu, J. Liaug and S. Adhikar, “Dynamic response of concrete pavement

structure with asphalt isolating layer under moving loads”, Journal of Traffic and

Transportation Engineering, vol. 1, no. 6, pp. 439-447, 2014.

93. P. C. Jorge, F. M. F. Simoes and A. P. Costa, “Dynamics of beams on non-

uniform nonlinear foundations subjected to moving loads”, Computers and


94. P. C. Jorge, F. M. F. Simoes and A. P. Costa, “Finite element dynamic analysis of

finite beams on a bilinear foundation under a moving load”, Journal of Sound and

Vibration, vol. 346, pp. 328-344, 2015.

Reference

163

95. R. Alebrahim, S. M. Haris, N. A. N. Mohamed and S. Abdullah, “Vibration

analysis of self-healing hybrid composite beam under moving mass”, Composite


96. H. Ozturk, Z. Kiral and B. G. Kiral, “Dynamic analysis of elastically supported

cracked beam subjected to a concentrated moving load”, Latin American Journal

of Solids and Structures”, vol. 13, no. 1, pp. 175-200, 2016.

97. C. Fu, “Dynamic behavior of a simply supported bridge with a switching crack

subjected to seismic excitations and moving trains”, Engineering Structures, vol.

110, pp. 59-69, 2016.

98. N. D. Beskou, S. V. Tsinopoulos and D. D. Theodorakopoulos, “Dynamic elastic

analysis of 3-Dflexible pavements under moving vehicles: A unified FEM

treatment”, Soil Dynamics and Earthquake Engineering, vol. 82, pp. 63-72, 2016.

99. A. Dutta and S. Talukdar, “Damage detection in bridges using accurate modal

parameters”, Finite Elements in Analysis and Design, vol. 40, pp. 287-304, 2004.

100. M. I. Friswell and J. E. T. Penny, “Crack modeling for structural health

monitoring”, Structural health monitoring, vol. 1, no. 2, pp. 139-148, 2002.

101. J. W. Lee, A. D. Kim, C. B. Yun, J. H. Yi and J. M. Shim, “Health-monitoring

method for bridges under ordinary traffic loadings”, Journal of Sound and

Vibration, vol. 257, no. 2, pp. 247-264, 2002.

102. M. A. B. Abdo and M. Hori, “A numerical study of structural damage detection

using changes in the rotation of mode shapes”, Journal of Sound and Vibration,

vol. 251, no. 2, pp. 227-239, 2002.

103. J. T. Kim and N. Stubbs, “Improved damage identification method based on

modal information”, Journal of Sound and Vibration, vol. 252, no. 2, pp. 223-

238, 2002.

104. T. G. Chondros, A. D. Dimarogonas and J. Yao, “A continuous cracked beam

vibration theory”, Journal of Sound and Vibration, vol. 215, no. 1, pp. 17-34,

1998.

105. T. G. Chondros and A. D. Dimarogonas, “Vibration of a cracked cantilever

beam”, Journal of Vibration and Acoustics, vol. 120, pp. 742-746, 1998.

106. F. Khoshnoudian and A. Eafandiari, “Structural damage diagnosis using modal

data”, Scientia Iranica A, vol. 18, no. 4, pp. 853-860, 2011.

107. A. S. J. Swamidas, X. Yang and R. Seshadri, “Identification of cracking in beam

structures using Timoshenko and Euler formulations”, Journal of Engineering

Mechanics, vol. 130, pp. 1297-1308, 2004.

108. D. F. Mazurek and J. T. DeWolf, “Experimental study of bridge monitoring

technique”, Journal of Structural Engineering, vol. 116, pp. 2532-2549, 1990.

109. R. Ruotolo and C. Surace, “Damage assessment of multiple cracked beams:

Numerical results and experimental validation”, Journal of Sound and Vibration,

vol. 206, no. 4, pp. 567-588, 1997.

Reference

164

110. A. K. Pandey and M. Biswas, “Damage detection in structures using the changes

in flexibility”, Journal of Sound and Vibration, vol. 169, no. 1, pp. 3-17, 1994.

111. T. D. Chaudhari and S. K. Maiti, “A study of vibration of geometrically

segmented beams with and without crack”, International Journal of Solids and


112. L. Majumder and C. S. Manohar, “A time-domain approach for damage detection

in beam structures using vibration data with a moving oscillator as an excitation

source”, Journal of Sound and Vibration, vol. 268, pp. 699-716, 2003.

113. S. Chinchalkar, “Determination of crack location in beams using natural

frequencies”, Journal of Sound and Vibration, vol. 247, no. 3, pp. 417-429, 2001.

114. N. Haritos and J. S. Owen, “The use of vibration data for damage detection in

bridges: A comparison of system identification and pattern recognition

approaches”, Structural Health Monitoring, vol. 3, no. 2, pp. 141-163, 2004.

115. H. Nahvi and M. Jabbari, “Crack detection in beams using experimental modal

data and finite element model”, International Journal of Mechanical Sciences,

vol. 47, pp. 1477-1497, 2005.

116. A. Alvandi and C. Cremona, “Assessment of vibration-based damage

identification techniques”, Journal of Sound and Vibration, vol. 292, pp. 179-

202, 2006.

117. S. Wang, Q. Ren and P. Qiao, “Structural damage detection using local damage

factor”, Journal of Vibration and Control, vol. 12, no. 9, pp. 955-973, 2006.

118. L. Fushun, L. Huajun, Y. Guangming, Z. Yantao, W. Weiying and S. Wanqing,

“New damage-locating method for bridges subjected to a moving load”, Journal

of Ocean University of China, vol. 6, no. 2, pp. 199-204, 2007.

119. L. Yu, H. Tommy and T. Chan, “Recent research on identification of moving

loads on bridges”, Journal of Sound and Vibration, vol. 305, pp. 3-21, 2007.

120. A. S. Sekhar, “Multiple cracks effects and identification”, Mechanical Systems

and Signal Processing, vol. 22, pp. 845-878, 2008.

121. N. Khaji, M. Shafiei and M. Jalalpour, “Closed-form solutions for crack detection

problem of Timoshenko beams with various boundary conditions”, International


122. J. Lee, “Identification of multiple cracks in a beam using vibration amplitudes”,

Journal of Sound and Vibration, vol. 326, pp. 205-212, 2009.

123. Z. Zhu, S. German and I. Brilakis, “Detection of large-scale concrete columns for

automated bridge inspection”, Automation in Construction, vol. 19, pp. 1047-

1055, 2010.

124. Q. Zhang, L. Jankowski and Z. Duan, “Simultaneous identification of moving

masses and structural damage”, Structural and Multidisciplinary Optimization,

vol. 42, pp. 907-922, 2010.

Reference

165

125. F. B. Sayyad and B. Kumar, “Identification of crack location and crack size in a

simply supported beam by measurement of natural frequencies”, Journal of

Vibration and Control, vol. 18, no. 2, pp. 183-190, 2010.

126. M. Dilena and A. Morassi, “Dynamic testing of a damaged bridge”, Mechanical

Systems and Signal Processing, vol. 25, pp. 1485-1507, 2011.

127. N. Roveri and A. Carcaterra, “Damage detection in structures under traveling

loads by Hilbert–Huang transform”, Mechanical Systems and Signal Processing,

vol. 28, pp. 128-144, 2012.

128. K. V. Nguyen, “Comparison studies of open and breathing crack detections of a

beam-like bridge subjected to a moving vehicle”, Engineering Structures, vol. 51,

pp. 306-314, 2013.

129. J. Li and S. S. Law, “Damage identification of a target substructure with moving

load excitation”, Mechanical Systems and Signal Processing, vol. 30, pp. 78-90,

2012.

130. S. Q. Zhu and S. S. Law, “Damage detection in simply supported concrete bridge

structure under moving vehicular loads”, Journal of Vibration and Acoustics, vol.

129, pp. 58-65, 2007.

131. N. T. Khiem and T. V. Lien, “Multi-crack detection for beam by the natural

frequencies”, Journal of Sound and Vibration, vol. 273, pp. 175-184, 2004.

132. D. P. Patil and S. K. Maiti, “Experimental verification of a method of detection of

multiple cracks in beams based on frequency measurements”, Journal of Sound

and Vibration, vol. 281, pp. 439-451, 2005.

133. V. Pakrashi, A. O’Connor and B. Basu, “A bridge-vehicle interaction based

experimental investigation of damage evolution”, Structural Health Monitoring,

vol. 9, no. 4, pp. 285-296, 2010.

134. Z. Zhou, B. Zhang, K. Xia, X. Li, G. Yan and K. Zhang, “Smart film for crack

monitoring of concrete bridges”, Structural Health Monitoring, vol. 10, no. 3, pp.

275-289, 2010.

135. A. S. Bouboulas and N. K. Anifantis, “Finite element modeling of a vibrating

beam with a breathing crack: observations on crack detection”, Structural Health

Monitoring, vol. 10, no. 2, pp. 131-145, 2011.

136. J. W. Zhan, H. Xia, S. Y. Chen and G. D. Roeck, “Structural damage

identification for railway bridges based on train-induced bridge responses and

sensitivity analysis”, Journal of Sound and Vibration, vol. 330, pp. 757-770,

2011.

137. A. Ghadami, A. Maghsoodi and H. R. Mirdamadi, “A new adaptable multiple-

crack detection algorithm in beam-like structures”, Archives of Mechanics, vol.

65, no. 6, pp. 469-483, 2013.

138. J. Hu and R. Y. Liang, “An integrated approach cracks using vibration

characteristics”, Journal of Franklin Institute, vol. 5, pp. 841-853, 1993.

Reference

166

139. B. Chomette, A. Fernandes and J. J. Sinou, “Cracks detection using active modal

damping and piezoelectric components”, Shock and Vibration, vol. 20, pp. 619-

631, 2013.

140. F. B. Nejad, A. Khorram and M. Rezaeian, “Analytical estimation of natural

frequencies and mode shapes of a beam having two cracks”, International


141. D. Wang, W. Xiang and H. Zhu, “Damage identification in beam type structures

based on statistical moment using a two step method”, Journal of Sound and

Vibration, vol. 333, pp. 745-760, 2014.

142. K. V. Nguyen, “Mode shapes analysis of a cracked beam and its application for

crack detection”, Journal of Sound and Vibration, vol. 333, pp. 848-872, 2014.

143. L. -T. Lee and J. Y. Wu, “Identification of crack locations and depths in a multi-

cracks beam by using local adaptive differential quadrature method”, Journal of

Process Mechanical Engineering, pp. 1-13, 2014.

144. P. Nandakumar and K. Shankar, “Multiple crack damage detection of structures

using the two crack transfer matrix”, Structural Health Monitoring, vol. 13, no. 5,

pp. 548-561, 2014.

145. N. T. Khiem and H. T. Tran, “A procedure for multiple crack identification in

beam-like structures from natural vibration mode”, Journal of Vibration and

Control, vol. 20, no. 9, pp. 1417-1427, 2014.

146. M. M. Ettefagh, H. Akbari, K. Asadi and F. Abbasi, “New structural damage-

identification method using modal updating and model reduction”, Journal of

Mechanical Engineering Science, pp. 1-19, 2014.

147. C. Schallhorn and S. Rahmatalla, “Crack detection and health monitoring of

highway steel-girder bridges”, Structural Health Monitoring, vol. 14, no. 3, pp.

281-299, 2015.

148. W. -Y. He and S. Zhu, “Moving load-induced response of damaged beam and its

application in damage localization”, Journal of Vibration and Control, pp. 1-17,

2015.

149. H. Zhu, L. Mao and S. Weng, “Calculation of dynamic response sensitivity to

substructural damage identification under moving load”, Advances in Structural


150. S. S. Law and Z. R. Lu, “Crack identification in beam from dynamic responses”,


151. S. S. Law and X. Q. Zhu, “Nonlinear characteristics of damaged concrete

structures under vehicular load”, Journal of Structural Engineering, vol. 131, no.

8, pp. 1277-1285, 2005.

152. J. Q. Bu, S. S. Law and X. Q. Zhu, “Innovative bridge condition assessment from

dynamic response of a passing vehicle”, Journal of Engineering Mechanics, vol.

132, pp. 1372-1379, 2006.

Reference

167

153. M. Karthikeyan, R. Tiwari and S. Talukdar, “Development of a technique to

locate and quantify a crack in a beam based on modal parameters”, Journal of

Vibration and Acoustics, vol. 129, pp. 390-395, 2007.

154. Z. Zhou, L. D. Wegner and B. F. Sparling, “Vibration-based detection of small-

scale damage on a bridge deck”, Journal of Structural Engineering, vol. 133, no.

9, pp. 1257-1267, 2007.

155. S. M. Al-said, “Crack detection in stepped beam carrying slowly moving mass”,

Journal of Vibration and Control, vol. 14, no. 12, pp. 1903-1920, 2008.

156. S. H. Yin and C. Y. Tang, “Identifying cable tension loss and deck damage in a

Cable-stayed bridge using a moving vehicle”, Journal of Vibration and Acoustics,

vol. 133, no. 2, 021007, 2011.

157. J. Li, S. S. Law and H. Hao, “Improved damage identification in bridge structures

subject to moving loads: Numerical and experimental studies”, International


158. J. Li and H. Hao, “Damage detection of shear connectors under moving loads

with relative displacement measurements”, Mechanical Systems and Signal

Processing, vol. 60, pp. 124-150, 2015.

159. X. Kong, C. S. Cai and B. Kong, “Damage detection based on transmissibility of

a vehicle and bridge coupled system”, Journal of Engineering Mechanics, vol.

141, no. 1, p. 04014102, 2015.

160. K. Aydin, “Vibratory characteristics of Euler-Bernoulli beams with an arbitrary

number of cracks subjected to axial load”, Journal of Vibration and Control, vol.

14, no. 4, pp. 485-510, 2008.

161. M. A. Dar, N. Subramanian, A. R. Dar and J. Raju, “Experimental investigations

on the structural behaviour of a distressed bridge”, Structural Engineering and

Mechanics, vol. 56, no. 4, pp. 695-705, 2015.

162. Y. Oshima, K. Yamamoto and K. Sugiura, “ Damage assessment of a bridge

based on mode shapes estimated by responses of passing vehicles”, Smart

Structures and Systems, vol. 13, no. 5, pp. 731-753, 2014.

163. K. Mazanoglu, “A novel methodology using simplified approaches for

identification of cracks in beams” Latin American Journal of Solids and


164. D. Feng and M. Feng, “Output-only damage detection using vehicle-induced

displacement response and mode shape curvature index”, Structural Control and

Health Monitoring, 2016, DOI: 10.1002/stc.

165. J. H. Chou and J. Ghaboussi, “Genetic algorithm in structural damage detection”,

Computers and Structures, vol. 79, pp. 1335-1357, 2001.

166. E. G. Shopova and N. G. V. Bancheva, “BASIC-A genetic algorithm for

engineering problems solution”, Computers and Chemical Engineering, vol. 30,

pp. 1293-1309, 2006.

Reference

168

167. X. He, M. Kawatani, T. Hayashikawa, H. Furuta and T. Matsumoto, “A bridge

damage detection approach using train-bridge interaction analysis and GA

optimization”, Procedia Engineering, vol. 14, pp. 769-776, 2011.

168. L. Sun, X. Cai and J. Yang, “Genetic algorithm-based optimum vehicle

suspension design using minimum dynamic pavement load as a design criterion”,


169. M. V. Baghmisheh, M. Peimani, M. H. Sadeghi and M. M. Ettefagh, “Crack

detection in beam-like structures using genetic algorithms”, Applied Soft

Computing, vol. 8, pp. 1150-1160, 2008.

170. V. Meruane and W. Heylen, “An hybrid real genetic algorithm to detect structural

damage using modal properties”, Mechanical Systems and Signal Processing,

vol. 25, pp. 1559-1573, 2011.

171. F. S. Buezas, M. B. Rosales and C. P. Filipich, “Damage detection with genetic

algorithms taking into account a crack contact model”, Engineering Fracture

Mechanics, vol. 78, pp. 695-712, 2011.

172. C. Na, S. P. Kim and H. G. Kwak, “Structural damage evaluation using genetic

algorithm”, Journal of Sound and Vibration, vol. 330, pp. 2772-2783, 2011.

173. M. Mehrjooa, N. Khaji and M. G. Ashtiany, “Application of genetic algorithm in

crack detection of beam-like structures using a new cracked Euler-Bernoulli

beam element”, Applied Soft Computing, vol. 13, pp. 867-880, 2013.

174. S. Y. Lee, “An advanced coupled genetic algorithm for identifying unknown

moving loads on bridge decks”, Mathematical Problems in Engineering, 2014,

Article ID 462341.

175. C. Chisari, C. Bedon and C. Amadio, “Dynamic and static identification of base-

isolated bridges using Genetic Algorithms”, Engineering Structures, vol. 102, pp.

80-92, 2015.

176. H. Alli, A. Ucar and Y. Demir, “The solutions of vibration control problems

using artificial neural networks”, Journal of the Franklin Institute, vol. 340, pp.

307-325, 2003.

177. X. Cao, Y. Sugiyama and Y. Mitsui, “Application of artificial neural networks to

load identification”, Computers and Structures, vol. 69, pp. 63-78, 1998.

178. M. A. Mahmoud and M. A. Kiefa, “Neural network solution of the inverse

vibration problem”, NDT&E International, vol. 32, pp. 91-99, 1999.

179. Z. Waszczyszyn and L. Ziemianski, “Neural networks in mechanics of structures

and materials-new results and prospects of application”, Computer and


180. C. Zang and M. Imregun, “Structural damage detection using artificial neural

networks and measured FRF data reduced via principal component projection”,

Journal of Sound and Vibration, vol. 242, no. 5, pp. 813-827, 2001.

Reference

169

181. S. W. Liu, J. H. Huang, J. C. Sung and C. C. Lee, “Detection of cracks using

neural networks and computational mechanics”, Computer Methods in Applied

Mechanics Engineering, vol. 191, pp. 2831-2845, 2002.

182. Q. Chen, Y. W. Chan and K. Worden, “Structural fault diagnosis and isolation

using neural networks based on response-only data”, Computers and Structures,

vol. 81, pp. 2165-2172, 2003.

183. C. Y. Kao and S. L. Hung, “Detection of structural damage via free vibration

responses generated by approximating artificial neural networks”, Computers and


184. M. Sahin and R. A. Shenoi, “Quantification and localisation of damage in beam-

like structures by using artificial neural networks with experimental validation”,

Engineering Structures, vol. 25, pp. 1785-1802, 2003.

185. S. Ataei, A. A. Aghakouchak, M. S. Marefat and S. Mohammadzadeh, “Sensor

fusion of a railway bridge load test using neural networks”, Expert Systems with

Applications, vol. 29, pp. 678-683, 2005.

186. J. Y. Kang, B. I. Choi, H. J. Lee, J. S. Kim and K. J. Kim, “Neural network

application in fatigue damage analysis under multiaxial random loadings”,

International Journal of Fatigue, vol. 28, pp. 132-140, 2006.

187. Y. J. Yan, L. Cheng, Z. Y. Wu and L. H. Yam, “Development in vibration-based

structural damage detection technique”, Mechanical Systems and Signal


188. Z. X. Li and X. M. Yang, “Damage identification for beams using ANN based on

statistical property of structural responses”, Computers and Structures, vol. 86,

pp. 64-71, 2008.

189. M. Mehrjooa, N. Khaji, H. Moharrami and A. Bahreininejad, “Damage detection

of truss bridge joints using artificial neural networks”, Expert Systems with

Applications, vol. 35, pp. 1122-1131, 2008.

190. N. Bakhary, H. Hao and A. J. Deeks, “Structure damage detection using neural

network with multi-stage substructuring”, Advances in Structural Engineering,

vol. 13, no. 1, pp. 1-16, 2010.

191. A. H. Al-Rahmani, H. A. Rasheed and Y. Najjar, “A combined soft computing-

mechanics approach to inversely predict damage in bridges”, Procedia Computer

Science, vol. 8, pp. 461-466, 2012.

192. J. Shu, Z. Zhang, I. Gonzalez and R. Karoumi, “The application of a damage

detection method using Artificial Neural Network and train-induced vibrations on

a simplified railway bridge model”, Engineering Structures, vol. 52, pp. 408-421,

2013.

193. M. Yaghinin, M. M. Khoshraftar and M. Fallahi, “A hybrid algorithm for

artificial neural network training”, Engineering Applications of Artificial

Intelligence, vol. 26, pp. 293-301, 2013.

Reference

170

194. A. A. Elshafey, N. Dawood, H. Marzouk and M. Haddara, “Predicting of crack

spacing for concrete by using neural networks”, Engineering Failure Analysis,

vol. 31, pp. 344-359, 2013.

195. S. Erkaya, “Analysis of the vibration characteristics of an experimental

mechanical system using neural networks”, Journal of Vibration and Control,

vol. 18, no. 13, pp. 2059-2072, 2011.

196. P. C. Pandey and S. V. Barai, “Multilayer perceptron in damage detection of

bridge structures”, Computer & Structures, vol. 54, no. 4, pp. 597-608, 1995.

197. P. C. Karninsk, “The approximate location of damage through the analysis of

natural frequencies with artificial neural networks”, Journal of Process

Mechanical Engineering, vol. 209, pp. 117-223, 1995.

198. A. Seibi and S. M. Al-Alawi, “Prediction of fracture toughness using artificial

neural networks (ANNs)”, Engineering Fracture Mechanics, vol. 56, no. 3, pp.

311-319, 1997.

199. J. Zhao, J. N. Ivan and J. T. DeWolf, “Structural damage detection using artificial

neural networks”, Journal of Infra Structure Systems, vol. 4, pp. 93-101, 1998.

200. T. Marwala and H. E. M. Hunt, “Fault identification using finite element models

and neural networks”, Mechanical Systems and Signal Processing, vol. 13, no. 3,

pp. 475-490, 1999.

201. C. C. Chang, T. Y. P. Chang, Y. G. Xu and M. L. Wang, “Structural damage

detection using an iterative neural network”, Journal of intelligent material

systems and structures, vol. 11, pp. 31-42, 2000.

202. T. K. Lin, C. C. J. Lin and K. C. Chang, “A neural network based methodology

for estimating bridge damage after major earthquakes”, Journal of the Chinese

Institute of Engineers, vol. 25, no. 4, pp. 415-424, 2002.

203. C. Zang, M. I. Friswell and M. Imregun, “Structural damage detection using

Independent component analysis”, Structural Health Monitoring, vol. 3, no. 1,

pp. 69-83, 2004.

204. K. Worden and J. M. Dulieu-Barton, “Damage detection in bridges using neural

networks for pattern recognition of vibration signatures”, Engineering Structures,

vol. 27, pp. 685-698, 2005.

205. J. J. Lee, J. W. Lee, J. H. Yi, C. B. Yun and H. Y. Jung, “Neural networks-based

damage detection for bridges considering errors in baseline finite element

models”, Journal of Sound and Vibration, vol. 280, pp. 555-578, 2005.

206. N. Bakhary, H. Hao and A. J. Deeks, “Damage detection using artificial neural

network with consideration of uncertainties”, Engineering Structures, vol. 29, pp.

2806-2815, 2007.

207. F. S. Wong, M. P. Thint and A. T. Tung, “On-line detection of structural damage

using neural networks”, Civil Engineering Systems, vol. 14, pp. 167-197, 2007.

Reference

171

208. P. M. Pawar, K. V. Reddy and R. Ganguli, “Damage detection in beams using

spatial Fourier analysis and neural networks”, Journal of Intelligent Material

Systems and Structures, vol. 18, pp. 347-359, 2007.

209. C. Gonzalez-Perez and J. Valdes-Gonzalez, “Identification of structural damage

in a vehicular bridge using artificial neural networks”, Structural Health


210. P. Li, “Structural damage localization using probabilistic neural networks”,

Mathematical and Computer Modelling, vol. 54, pp. 965-969, 2011.

211. D. R. Parhi and A. K. Dash, “Application of neural networks and finite elements

for condition monitoring of structures”, Mechanical Engineering Science, vol.

225, pp. 1329-1339, 2011.

212. A. A. Elshafey, N. Dawood, H. Marzouk and M. Haddara, “Crack width in

concrete using artificial neural networks”, Engineering Structures, vol. 52, pp.

676-686, 2013.

213. O. Hasancebi and T. Dumlupınar, “Detailed load rating analyses of bridge

populations using nonlinear finite element models and artificial neural networks”,

Computers and Structures, vol. 128, pp. 48-63, 2013.

214. R. P. Bandara, H. T. T. Chan and P. D. Thambiratnam, “Structural damage

detection method using frequency response functions”, Structural Health


215. K. Aydin and O. Kisi, “Damage diagnosis in beam-like structures by artificial

neural networks”, Journal of Civil Engineering and Management, vol. 21, no. 5,

pp. 591-604, 2015.

216. S. S. Kourehli, “Damage assessment in structures using incomplete modal data

and artificial neural network”, International Journal of Structural Stability and

Dynamics, vol. 15, no. 6, p. 1450087, 2015.

217. H. A. Alavi, H. Hasni, N. Lajnef, K. Chatti and F. Faridazar, “An intelligent

structural damage detection approach based on self-powered wireless sensor

data”, Automation in Construction, vol. 62, pp. 24-44, 2016.

218. G. Yu, H. Qiu, D. Djurdjanovic and J. Lee, “Feature signature prediction of a

boring process using neural network modeling with confidence bounds”,

International Journal of Advance Manufacturing Technology, vol. 30, no. 7, pp.

614-621, 2006.

219. S. Y. Xiong and J. P. Withers, “An evaluation of recurrent neural network

modelling for the prediction of damage evolution during forming”, Journal of

Materials Processing Technology, vol. 170, pp. 551-562, 2005.

220. S. Ekici, S. Yildirim and M. Poyraz, “A transmission line fault locator based on

Elman recurrent networks”, Applied Soft Computing, vol. 9, pp. 341-347, 2009.

Reference

172

221. M. M. Abdelhameed and H. Darabi, “Neural network based design of fault-

tolerant controllers for automated sequential manufacturing systems”,

Mechatronics, vol. 19, pp. 705-714, 2009.

222. T. J. Connor, R. D. Martin and L. E. Atlas, “Recurrent neural networks and

robust time series prediction”, IEEE Transactions on Neural Networks, vol. 5, no.

2, pp. 240-254, 1994.

223. B. A. Pearlmutter, “Gradient calculations for dynamic recurrent neural networks:

A survey”, IEEE Transactions on Neural Networks, vol. 6, no. 5, pp. 1212-1228,

1995.

224. P. W. Tse and D. P. Atherton, “Prediction of machine deterioration using

vibration based fault trends and recurrent neural networks”, Journal of Vibration

and Acoustics, vol. 121, pp. 355-362, 1999.

225. C. G. Gan and D. Danai, “Model based recurrent neural network for modeling

nonlinear dynamic system”, IEEE Transactions of Systems, Man and

Cybernetics, vol. 30, no. 2, pp. 344-351, 2000.

226. Z. Waszczyszyn and L. Ziemianski, “Neural networks in mechanics of structures

and materials-new results and prospects applications, Computers and Structures,

vol. 79, pp. 2261-2276, 2001.

227. M. T. Valoor, K. Chandrasekhara and S. Agarwal, “Self-adaptive vibration

control of smart composite beams using recurrent neural network architectures”,

International Journal of Solids and Structures, vol. 38, pp. 7857-7874, 2001.

228. S. Seker, E. Ayaz and E. Turkcan, “Elman’s recurrent neural network

applications to condition monitoring in nuclear power plant and rotating

machinery”, Engineering Applications of Artificial Intelligence, vol. 16, pp. 647-

656, 2003.

229. A. M. Schafer and H. G. Zimmermann, “Recurrent neural networks are universal

approximators”, International Journal of Neural Systems, vol. 17, no. 4, pp. 253-

263, 2007.

230. A. Thammano and P. Ruxpakawong, “Nonlinear dynamic systems identification

using recurrent neural with multi-segment piecewise-linear connection weight”,

Memetic Computing, vol. 2, pp. 273-282, 2010.

231. R. Coban, “A context layered locally recurrent neural network for dynamic

system identification”, Engineering Applications of Artificial Intelligence, vol.

26, pp. 241-250, 2013.

232. H. Sohn, J. A. Czarnecki and C. R. Farrar, “ Structural health monitoring using

statistical process control”, Journal of Structural Engineering, vol. 126, no. 11,

pp. 1356-1363, 2000.

233. M. L. Fugate, H. Sohn and C. Farrar, “Vibration-based damage detection using

statistical process control”, Mechanical Systems and Signal Processing, vol. 15,

no. 4, pp. 707-721, 2001.

Reference

173

234. Y. Lei, A. S. Kiremidjian, K. K. Nair, J. P. Lynch, K. H. Law, T. W. Kenny, E.

Carryer and V. Kottapalli, “Statistical damage detection using time series analysis

on a structural health monitoring benchmark problem”, Proceedings of the 9th

International Conference on Applications of Statistics and Probability in Civil

Engineering, San Francisco, pp. 6-9, 2003.

235. Y. Lu and F. Gao, “A novel time-domain auto-regressive model for structural

damage diagnosis”, Journal of Sound and Vibration, vol. 283, pp. 1031-1049,

2005.

236. S. G. Mattson and S. M. Pandit, “Statistical moments of autoregressive model

residuals for damage localisation”, Mechanical Systems and Signal Processing,

vol. 20, pp. 627-645, 2006.

237. J. Zhang, Y. L. Xu, Y. Xia and J. Li, “A new statistical moment-based structural

damage detection method”, Structural Engineering and Mechanics, vol. 30, no. 4,

pp. 445-466, 2008.

238. M. Gul and F. N. Catbas, “Statistical pattern recognition for Structural Health

Monitoring using time series modeling: Theory and experimental verifications”,

Mechanical Systems and Signal Processing, vol. 23, pp. 2192-2204, 2009.

239. S. Law and J. Li, “Updating the reliability of a concrete bridge structure based on

condition assessment with uncertainties”, Engineering Structures, vol. 32, pp.

286-296, 2010.

240. J. L. Zapico-Valle, M. Garcia-Dieguez, M. P. Gonzalez-Martınez and K. Worden,

“Experimental validation of a new statistical process control feature for damage

detection”, Mechanical Systems and Signal Processing, vol. 25, pp. 2513-2525,

2011.

241. K. Kwon and D. M. Frangopol, “Bridge fatigue assessment and management

using reliability-based crack growth and probability of detection models”,

Probabilistic Engineering Mechanics, vol. 26, pp. 471-480, 2011.

242. A. A. Mosavi, D. S. Dickey and S. Rizkalla, “ Identifying damage locations under

ambient vibrations utilizing vector autoregressive models and Mahalanobis

distances”, Mechanical System and Signal Processing, vol. 26, pp. 254-267,

2012.

243. F. Cavadas, I. F. C. Smith and J. Figueiras, “Damage detection using data-driven

methods applied to moving-load responses”, Mechanical Systems and Signal


244. F. P. Kopsaftopoulos and S. D. Fassois, “A functional model based statistical

time series method for vibration based damage detection, localization, and

magnitude estimation”, Mechanical Systems and Signal Processing, vol. 39, pp.

143-161, 2013.

Reference

174

245. B. Phares, P. Lu, T. Wipf, L. Greimann and J. Seo, “Field validation of a

statistical-based bridge damage-detection algorithm”, Journal of Bridge


246. D. Wang, W. Xiang and H. Zhu, “ Damage identification in beam type structures

based on statistical moment using two step method”, Journal of Sound and

Vibration, vol. 333, pp. 745-760, 2014.

247. L. Yu and J. Zhu, “Nonlinear damage detection using higher statistical moments

of structural responses”, Structural Engineering and Mechanics, vol. 54, no. 2,

pp. 221-237, 2015.

248. A. J. Reiff, M. Sanayei and R. M. Vogel, “Statistical bridge damage detection

using girder distribution factors”, Engineering Structures, vol. 109, pp. 139-151,

2016.

249. M. G. Masciottaa, L. F. Ramos, P. B. Lourenco, M. Vastab and G. D. Roeck, “A

spectrum-driven damage identification technique: Application and validation

through the numerical simulation of the Z24 Bridge”, Mechanical Systems and

Signal Processing, vol. 70, pp. 578-600, 2016.

250. M. Dilenaa, M. P. Limongelli and A. Morassi, “Damage localization in bridges

via the FRF interpolation method”, Mechanical Systems and Signal Processing,

vol. 52, pp. 162-180, 2015.

251. H. Gokdag, “A crack identification method for beam type structures subject to

moving vehicle using particle swarm optimization”, Gazi University Journal of

Science, vol. 26, no. 3, pp. 439-448, 2013.

252. F. Kang, J. Li and Q. Xu, “Damage detection based on improved particle swarm

optimization using vibration data”, Applied Soft Computing, vol. 12, pp. 2329-

2335, 2012.

253. D. Hester and A. Gonzalez, “A wavelet-based damage detection algorithm based

on bridge acceleration response to a vehicle”, Mechanical Systems and Signal


254. B. Sahoo and D. Maity, “Damage assessment of structures using hybrid neuro-

genetic algorithm”, Applied Soft Computing, vol. 7, pp. 89-104, 2007.

255. G. Li, S. He, Y. Ju and K. Du, “Long distance precision inspection method for

bridge cracks with image processing”, Automation in Construction, vol. 41, pp.

83-95, 2014.

256. R. S. Adhikari, O. Moselhi and A. Bagchi, “Image-based retrieval of concrete

crack properties for bridge inspection”, Automation in Construction, vol. 39, pp.

180-194, 2014.

257. J. Li and H. Hao, “Health monitoring of joint in steel truss bridges with relative

displacement sensors”, Measurement, vol. 88, pp. 360-371, 2016.

258. W. T. Thomson, “Theory of vibration with application”, Third Edition, CBS

Publishers, New Delhi, India.

Reference

175

259. L. Fryba, “Vibration of solids and structures”, Third Edition, Thomas Telford

Ltd, Prague, 2002.

260. M. Paz, “Structural Dynamics: Theory and Computation”, Third Edition, An

International Thomson Publishing Company, New York.

261. A. K. Chopra, “Dynamics of structures: Theory and application to earthquake

engineering”, Fourth Edition, Prentice-Hall International Series, New York.

262. D. C. Montgomery, “Introduction to statistical control”, Fifth edition, Wiley

Publications.

263. P. J. Brockwell and R. A. Davis, “Time series: Theory and methods”, Second

edition, Springer, 1999.

264. G. E. P. Box, G. M. Jenkins and G. C. Reinsel, “Time series analysis: Forecasting

and control”, Third edition, Prentice Hall International, 1994.

265. H. Demuth and M. Beale, “Neural Networks Toolbox”, The Math Works, Inc

Natick, 2002.

266. H. Yu and B. M. Wilamski, “Levenberg-Marquardt Training, Lecture notes on

intelligent systems, Auburn University.

176

Dissemination

Publications:

International Journals

1) Vibrational Analysis of Structures: A Review, Dayal. R Parhi and Shakti. P Jena,

International Journal of Applied Artificial Intelligence Engineering Systems, vol. 4, no. 2,

pp.83-94, 2012.

2) Dynamic Response of Cracked Simple Beam Carrying Moving Mass, Shakti. P Jena and

Dayal. R. Parhi, International Journal of Applied Engineering Research, vol. 9, no. 26,

pp.8794-8797, 2014. (Scopus)

3) Dynamic Deflection of a Cantilever Beam Carrying Moving Mass, Shakti. P Jena and

Dayal. R. Parhi, Applied Mechanics and Materials, vols. 592-594, pp. 1040-1044, 2014.

(Scopus)

4) Dynamic Response of Damaged Cantilever Beam Subjected to Traversing Mass,

International Journal for Technological Research in Engineering, vol. 2, no. 7, pp. 860-86,

2015.

5) Parametric Study on the Response of Cracked Structure Subjected to Moving Mass,

Shakti. P Jena and Dayal. R. Parhi, Journal of Vibration Engineering and Technology

(Accepted to be published in vol. 5, no. 1, 2017. (SCI)

6) Comparative Study on Cracked Beam with Different types of cracks Carrying Moving

Mass, Shakti. P Jena, Dayal R. Parhi and Devasis Mishra, Structural Engineering and

Mechanics, International Journal, vol. 56, no. 5, pp.797-811, 2015. (SCI)

7) Dynamic and experimental analysis on response of multiple cracked structures carrying

transit mass, Dayal. R Parhi and Shakti. P Jena, Journal of Risk and Reliability, SCI

(Accepted)

8) Dynamic Response and Analysis of Damaged Beam Structure Subjected to Traversing

Mass, Shakti. P Jena and Dayal R. Parhi, Journal of Steel and Composite Structures.

(Under review since 8th April 2016, SCI)

9) Response Analysis of Cracked Structure Subjected to Transit Mass- A Parametric

Study, Shakti. P Jena and Dayal R. Parhi, Journal of vibroengineering. (Under review

since 18th April 2016, SCI)

10) Response of structure to high speed mass, Shakti. P Jena and Dayal R. Parhi, Procedia

engineering, vol. 144, pp.1435-1442, 2016. (Scopus)

Dissemination

177

International Conferences

1) Analytical and Computational Study for the Dynamic Response of a Cantilever Beam

Carrying Moving Mass, Shakti. P Jena and Dayal R. Parhi, International Conferences in

Smart Technologies for Mechanical Engineering, 25-26 October, 2013, Delhi

Technological University, Delhi.

2) Dynamic Analysis of Cantilever beam with Moving Mass, Shakti. P Jena and Dayal R.

Parhi, International Conference on Structural Engineering and Mechanics, 20-22

December, 2013, NIT, Rourkela, Odisha.

3) Dynamic response of a simply supported beam with traversing mass, Shakti. P Jena and

Dayal R. Parhi, International Conference on Industrial Engineering Science and

Applications, 2-4 April, 2014, NIT, Durgapur, Westbengal.

4) Numerical Analysis for the Dynamic Analysis of Cantilever Beam Carrying Moving Load,

Shakti. P Jena and Dayal R. Parhi, Sixth International Conference on Theoretical,

Applied, Computational and Experimental Mechanics, 29-31 December, 2014, IIT,

Kharagpur, Westbengal.

5) Response of Cracked Cantilever Beam Subjected to Traversing Mass, Shakti. P Jena,

Dayal R. Parhi and Devasis Mishra, 2-3 December, 2015, ASME India Gas Turbine

Conference, HICC, Hyderabad, Andhrapradesh.

178

Vitae

Mr Shakti Prasanna Jena was born in 1st

July, 1984 in Jajpur, Odisha. He has completed

his +2 Science education from Kendrapara College (Kendrapara), in 2001 and Bachelor of

Engineering in Mechanical Engineering from Orissa Engineering College, (B.P.U.T.

University) in 2007. He got his Master of Technology in Mechanical Engineering

(Mechanical System Design) from B.P.U.T (Rourkela) in 2011. After completion of his

Post-Graduation, he joined the Ph.D. program at Department of Mechanical Engineering,

National Institute of Technology, Rourkela, in July 2012 and submitted her Ph.D. thesis in

July 2016. His research interest includes mechanical vibration, structural dynamics,

damage analysis in cracked structure and various soft computing methods etc.

dynamic analysis and fault detection of...

Documents